chapter 7 resource masters - ktl math classes...©glencoe/mcgraw-hill iv glencoe algebra 2...
TRANSCRIPT
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9
ANSWERS FOR WORKBOOKS The answers for Chapter 7 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-828010-9 Algebra 2Chapter 7 Resource Masters
1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02
Glencoe/McGraw-Hill
© Glencoe/McGraw-Hill iii Glencoe Algebra 2
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Lesson 7-1Study Guide and Intervention . . . . . . . . 375–376Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 377Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 378Reading to Learn Mathematics . . . . . . . . . . 379Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 380
Lesson 7-2Study Guide and Intervention . . . . . . . . 381–382Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 383Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 384Reading to Learn Mathematics . . . . . . . . . . 385Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 386
Lesson 7-3Study Guide and Intervention . . . . . . . . 387–388Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 389Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 390Reading to Learn Mathematics . . . . . . . . . . 391Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 392
Lesson 7-4Study Guide and Intervention . . . . . . . . 393–394Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 395Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 396Reading to Learn Mathematics . . . . . . . . . . 397Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 398
Lesson 7-5Study Guide and Intervention . . . . . . . 399–400Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 401Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 402Reading to Learn Mathematics . . . . . . . . . . 403Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 404
Lesson 7-6Study Guide and Intervention . . . . . . . . 405–406Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 407Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 408Reading to Learn Mathematics . . . . . . . . . . 409Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 410
Lesson 7-7Study Guide and Intervention . . . . . . . . 411–412Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 413Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 414Reading to Learn Mathematics . . . . . . . . . . 415Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 416
Lesson 7-8Study Guide and Intervention . . . . . . . . 417–418Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 419Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 420Reading to Learn Mathematics . . . . . . . . . . 421Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 422
Lesson 7-9Study Guide and Intervention . . . . . . . . 423–424Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 425Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 426Reading to Learn Mathematics . . . . . . . . . . 427Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 428
Chapter 7 AssessmentChapter 7 Test, Form 1 . . . . . . . . . . . . 429–430Chapter 7 Test, Form 2A . . . . . . . . . . . 431–432Chapter 7 Test, Form 2B . . . . . . . . . . . 433–434Chapter 7 Test, Form 2C . . . . . . . . . . . 435–436Chapter 7 Test, Form 2D . . . . . . . . . . . 437–438Chapter 7 Test, Form 3 . . . . . . . . . . . . 439–440Chapter 7 Open-Ended Assessment . . . . . . 441Chapter 7 Vocabulary Test/Review . . . . . . . 442Chapter 7 Quizzes 1 & 2 . . . . . . . . . . . . . . . 443Chapter 7 Quizzes 3 & 4 . . . . . . . . . . . . . . . 444Chapter 7 Mid-Chapter Test . . . . . . . . . . . . 445Chapter 7 Cumulative Review . . . . . . . . . . . 446Chapter 7 Standardized Test Practice . . 447–448Unit 2 Test/Review (Ch. 5–7) . . . . . . . . 449–450First Semester Test (Ch. 1–7) . . . . . . . 451–452
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A40
© Glencoe/McGraw-Hill iv Glencoe Algebra 2
Teacher’s Guide to Using theChapter 7 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 7 Resource Masters includes the core materials neededfor Chapter 7. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 7-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.
Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
© Glencoe/McGraw-Hill v Glencoe Algebra 2
Assessment OptionsThe assessment masters in the Chapter 7Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 406–407. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
77
© Glencoe/McGraw-Hill vii Glencoe Algebra 2
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 7.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
composition of functions
depressed polynomial
end behavior
Factor Theorem
Fundamental Theorem of Algebra
inverse function
inverse relation
leading coefficients
location principle
one-to-one
(continued on the next page)
© Glencoe/McGraw-Hill viii Glencoe Algebra 2
Vocabulary Term Found on Page Definition/Description/Example
polynomial function
polynomial in one variable
power function
quadratic form
Rational Zero Theorem
relative maximum
relative minimum
remainder theorem
square root function
synthetic substitution
sihn·THEH·tihk
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
77
Study Guide and InterventionPolynomial Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
© Glencoe/McGraw-Hill 375 Glencoe Algebra 2
Less
on
7-1
Polynomial Functions
A polynomial of degree n in one variable x is an expression of the form
Polynomial in a0xn � a1xn � 1 � … � an � 2x2 � an � 1x � an,One Variable where the coefficients a0, a1, a2, …, an represent real numbers, a0 is not zero,
and n represents a nonnegative integer.
The degree of a polynomial in one variable is the greatest exponent of its variable. Theleading coefficient is the coefficient of the term with the highest degree.
A polynomial function of degree n can be described by an equation of the form
Polynomial P(x ) � a0xn � a1xn � 1 � … � an � 2x2 � an � 1x � an,Function where the coefficients a0, a1, a2, …, an represent real numbers, a0 is not zero,
and n represents a nonnegative integer.
What are the degree and leading coefficient of 3x2 � 2x4 � 7 � x3?Rewrite the expression so the powers of x are in decreasing order.�2x4 � x3 � 3x2 � 7This is a polynomial in one variable. The degree is 4, and the leading coefficient is �2.
Find f(�5) if f(x) � x3 � 2x2 � 10x � 20.f(x) � x3 � 2x2 � 10x � 20 Original function
f(�5) � (�5)3 � 2(�5)2 � 10(�5) � 20 Replace x with �5.
� �125 � 50 � 50 � 20 Evaluate.
� �5 Simplify.
Find g(a2 � 1) if g(x) � x2 � 3x � 4.g(x) � x2 � 3x � 4 Original function
g(a2 � 1) � (a2 � 1)2 � 3(a2 � 1) � 4 Replace x with a2 � 1.
� a4 � 2a2 � 1 � 3a2 � 3 � 4 Evaluate.
� a4 � a2 � 6 Simplify.
State the degree and leading coefficient of each polynomial in one variable. If it isnot a polynomial in one variable, explain why. 8; 81. 3x4 � 6x3 � x2 � 12 4; 3 2. 100 � 5x3 � 10x7 7; 10 3. 4x6 � 6x4 � 8x8 � 10x2 � 20
4. 4x2 � 3xy � 16y2 5. 8x3 � 9x5 � 4x2 � 36 6. � � �not a polynomial in 5; �9one variable; contains 6; �two variables
Find f(2) and f(�5) for each function.
7. f(x) � x2 � 9 8. f(x) � 4x3 � 3x2 � 2x � 1 9. f(x) � 9x3 � 4x2 � 5x � 7�5; 16 23; �586 73; �1243
1�
1�72
x3�36
x6�25
x2�18
Example 1Example 1
Example 2Example 2
Example 3Example 3
ExercisesExercises
© Glencoe/McGraw-Hill 376 Glencoe Algebra 2
Graphs of Polynomial Functions
If the degree is even and the leading coefficient is positive, thenf(x) → �� as x → ��
f(x) → �� as x → ��
If the degree is even and the leading coefficient is negative, then
End Behaviorf(x) → �� as x → ��
of Polynomialf(x) → �� as x → ��
FunctionsIf the degree is odd and the leading coefficient is positive, then
f(x) → �� as x → ��
f(x) → �� as x → ��
If the degree is odd and the leading coefficient is negative, thenf(x) → �� as x → ��
f(x) → �� as x → ��
Real Zeros ofThe maximum number of zeros of a polynomial function is equal to the degree of the polynomial.
a PolynomialA zero of a function is a point at which the graph intersects the x-axis.
FunctionOn a graph, count the number of real zeros of the function by counting the number of times thegraph crosses or touches the x-axis.
Determine whether the graph represents an odd-degree polynomialor an even-degree polynomial. Then state the number of real zeros.
As x → ��, f(x) → �� and as x → ��, f(x) → ��,so it is an odd-degree polynomial function.The graph intersects the x-axis at 1 point,so the function has 1 real zero.
Determine whether each graph represents an odd-degree polynomial or an even-degree polynomial. Then state the number of real zeros.
1. 2. 3.
even; 6 even; 1 double zero odd; 3
x
f(x)
Ox
f(x)
Ox
f(x)
O
x
f(x)
O
Study Guide and Intervention (continued)
Polynomial Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
ExampleExample
ExercisesExercises
Skills PracticePolynomial Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
© Glencoe/McGraw-Hill 377 Glencoe Algebra 2
Less
on
7-1
State the degree and leading coefficient of each polynomial in one variable. If it isnot a polynomial in one variable, explain why.
1. a � 8 1; 1 2. (2x � 1)(4x2 � 3) 3; 8
3. �5x5 � 3x3 � 8 5; �5 4. 18 � 3y � 5y2 � y5 � 7y6 6; 7
5. u3 � 4u2v2 � v4 6. 2r � r2 �
No, this polynomial contains two No, this is not a polynomialbecause
variables, u and v. �r12� cannot be written in the form rn,
where n is a nonnegative integer.
Find p(�1) and p(2) for each function.
7. p(x) � 4 � 3x 7; �2 8. p(x) � 3x � x2 �2; 10
9. p(x) � 2x2 � 4x � 1 7; 1 10. p(x) � �2x3 � 5x � 3 0; �3
11. p(x) � x4 � 8x2 � 10 �1; 38 12. p(x) � �13�x2 � �
23�x � 2 3; 2
If p(x) � 4x2 � 3 and r(x) � 1 � 3x, find each value.
13. p(a) 4a2 � 3 14. r(2a) 1 � 6a
15. 3r(a) 3 � 9a 16. �4p(a) �16a2 � 12
17. p(a2) 4a4 � 3 18. r(x � 2) 7 � 3x
For each graph,a. describe the end behavior,b. determine whether it represents an odd-degree or an even-degree polynomial
function, andc. state the number of real zeroes.
19. 20. 21.
f(x) → �� as x → ��, f(x) → �� as x → ��, f(x) → �� as x → ��,f(x) → �� as x → ��; f(x) → �� as x → ��; f(x) → �� as x → ��;
x
f(x)
Ox
f(x)
Ox
f(x)
O
1�r2
© Glencoe/McGraw-Hill 378 Glencoe Algebra 2
State the degree and leading coefficient of each polynomial in one variable. If it isnot a polynomial in one variable, explain why.
1. (3x2 � 1)(2x2 � 9) 4; 6 2. �15�a3 � �
35�a2 � �
45�a 3; �
15
�
3. � 3m � 12 Not a polynomial; 4. 27 � 3xy3 � 12x2y2 � 10y
�m2
2� cannot be written in the form No, this polynomial contains two
mn for a nonnegative integer n. variables, x and y.
Find p(�2) and p(3) for each function.
5. p(x) � x3 � x5 6. p(x) � �7x2 � 5x � 9 7. p(x) � �x5 � 4x3
24; �216 �29; �39 0; �135
8. p(x) � 3x3 � x2 � 2x � 5 9. p(x) � x4 � �12�x3 � �
12�x 10. p(x) � �
13�x3 � �
23�x2 � 3x
�37; 73 13; 93 �6; 24
If p(x) � 3x2 � 4 and r(x) � 2x2 � 5x � 1, find each value.
11. p(8a) 12. r(a2) 13. �5r(2a) 192a2 � 4 2a4 � 5a2 � 1 �40a2 � 50a � 5
14. r(x � 2) 15. p(x2 � 1) 16. 5[p(x � 2)]2x2 � 3x � 1 3x4 � 6x2 � 1 15x2 � 60x � 40
For each graph,a. describe the end behavior,b. determine whether it represents an odd-degree or an even-degree polynomial
function, andc. state the number of real zeroes.
17. 18. 19.
f(x) → �� as x → ��, f(x) → �� as x → ��, f(x) → �� as x → ��,f(x) → �� as x → ��; f(x) → �� as x → ��; f(x) → �� as x → ��;even; 2 even; 1 odd; 5
20. WIND CHILL The function C(s) � 0.013s2 � s � 7 estimates the wind chill temperatureC(s) at 0�F for wind speeds s from 5 to 30 miles per hour. Estimate the wind chilltemperature at 0�F if the wind speed is 20 miles per hour. about �22�F
x
f(x)
Ox
f(x)
Ox
f(x)
O
2�m2
Practice (Average)
Polynomial Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
Reading to Learn MathematicsPolynomial Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
© Glencoe/McGraw-Hill 379 Glencoe Algebra 2
Less
on
7-1
Pre-Activity Where are polynomial functions found in nature?
Read the introduction to Lesson 7-1 at the top of page 346 in your textbook.
• In the honeycomb cross section shown in your textbook, there is 1 hexagonin the center, 6 hexagons in the second ring, and 12 hexagons in the thirdring. How many hexagons will there be in the fourth, fifth, and sixth rings?18; 24; 30
• There is 1 hexagon in a honeycomb with 1 ring. There are 7 hexagons ina honeycomb with 2 rings. How many hexagons are there in honeycombswith 3 rings, 4 rings, 5 rings, and 6 rings?19; 37; 61; 91
Reading the Lesson
1. Give the degree and leading coefficient of each polynomial in one variable.
degree leading coefficient
a. 10x3 � 3x2 � x � 7
b. 7y2 � 2y5 � y � 4y3
c. 100
2. Match each description of a polynomial function from the list on the left with thecorresponding end behavior from the list on the right.
a. even degree, negative leading coefficient iii i. f(x) → �� as x → ��;f(x) → �� as x → ��
b. odd degree, positive leading coefficient iv ii. f(x) → �� as x → ��;f(x) → �� as x → ��
c. odd degree, negative leading coefficient ii iii. f(x) → �� as x → ��;f(x) → �� as x → ��
d. even degree, positive leading coefficient i iv. f(x) → �� as x → ��;f(x) → �� as x → ��
Helping You Remember
3. What is an easy way to remember the difference between the end behavior of the graphsof even-degree and odd-degree polynomial functions?
Sample answer: Both ends of the graph of an even-degree functioneventually keep going in the same direction. For odd-degree functions,the two ends eventually head in opposite directions, one upward, theother downward.
1000
�25103
© Glencoe/McGraw-Hill 380 Glencoe Algebra 2
Approximation by Means of PolynomialsMany scientific experiments produce pairs of numbers [x, f(x)] that can be related by a formula. If the pairs form a function, you can fit a polynomial to the pairs in exactly one way. Consider the pairs given by the following table.
We will assume the polynomial is of degree three. Substitute the given values into this expression.
f(x) � A � B(x � x0) � C(x � x0)(x � x1) � D(x � x0)(x � x1)(x � x2)
You will get the system of equations shown below. You can solve this system and use the values for A, B, C, and D to find the desired polynomial.
6 � A11 � A � B(2 � 1) � A � B39 � A � B(4 � 1) � C(4 � 1)(4 � 2) � A � 3B � 6C
�54 � A � B(7 � 1) � C(7 � 1)(7 � 2) � D(7 � 1)(7 � 2)(7 � 4) � A � 6B � 30C � 90D
Solve.
1. Solve the system of equations for the values A, B, C, and D.
2. Find the polynomial that represents the four ordered pairs. Write your answer in the form y � a � bx � cx2 � dx3.
3. Find the polynomial that gives the following values.
4. A scientist measured the volume f(x) of carbon dioxide gas that can be absorbed by one cubic centimeter of charcoal at pressure x. Find the values for A, B, C, and D.
x 120 340 534 698
f (x) 3.1 5.5 7.1 8.3
x 8 12 15 20
f (x) �207 169 976 3801
x 1 2 4 7
f (x) 6 11 39 �54
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
Study Guide and InterventionGraphing Polynomial Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
© Glencoe/McGraw-Hill 381 Glencoe Algebra 2
Less
on
7-2
Graph Polynomial Functions
Location PrincipleSuppose y � f(x) represents a polynomial function and a and b are two numbers such thatf(a) � 0 and f(b) � 0. Then the function has at least one real zero between a and b.
Determine the values of x between which each real zero of thefunction f(x) � 2x4 � x3 � 5 is located. Then draw the graph.Make a table of values. Look at the values of f(x) to locate the zeros. Then use the points tosketch a graph of the function.
The changes in sign indicate that there are zerosbetween x � �2 and x � �1 and between x � 1 andx � 2.
Graph each function by making a table of values. Determine the values of x atwhich or between which each real zero is located.
1. f(x) � x3 � 2x2 � 1 2. f(x) � x4 � 2x3 � 5 3. f(x) � �x4 � 2x2 � 1
between 0 and �1; between �2 and �3; at �1 at 1; between 1 and 2 between 1 and 2
4. f(x) � x3 � 3x2 � 4 5. f(x) � 3x3 � 2x � 1 6. f(x) � x4 � 3x3 � 1
at �1, 2 between 0 and 1 between 0 and 1;between 2 and 3
x
f(x)
Ox
f(x)
Ox
f(x)
O
x
f(x)
Ox
f(x)
O
x
f(x)
O 4 8–4–8
8
4
–4
–8
x
f(x)
O
x f(x)
�2 35
�1 �2
0 �5
1 �4
2 19
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 382 Glencoe Algebra 2
Maximum and Minimum Points A quadratic function has either a maximum or aminimum point on its graph. For higher degree polynomial functions, you can find turningpoints, which represent relative maximum or relative minimum points.
Graph f(x) � x3 � 6x2 � 3. Estimate the x-coordinates at which therelative maxima and minima occur.Make a table of values and graph the function.
A relative maximum occursat x � �4 and a relativeminimum occurs at x � 0.
Graph each function by making a table of values. Estimate the x-coordinates atwhich the relative maxima and minima occur.
1. f(x) � x3 � 3x2 2. f(x) � 2x3 � x2 � 3x 3. f(x) � 2x3 � 3x � 2
max. at 0, min. at 2 max. about �1, max. about �1, min. about 0.5 min. about 1
4. f(x) � x4 � 7x � 3 5. f(x) � x5 � 2x2 � 2 6. f(x) � x3 � 2x2 � 3
min. about 1 max. at 0, max. about �1, min. about 1 min. at 0
x
f(x)
Ox
f(x)
Ox
f(x)
O 4 8–4–8
8
4
–4
–8
x
f(x)
Ox
f(x)
Ox
f(x)
O
x
f(x)
O2–2–4
24
16
8
← indicates a relative maximum
← zero between x � �1, x � 0
← indicates a relative minimum
x f(x)
�5 22
�4 29
�3 24
�2 13
�1 2
0 �3
1 4
2 29
Study Guide and Intervention (continued)
Graphing Polynomial Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
ExampleExample
ExercisesExercises
Skills PracticeGraphing Polynomial Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
© Glencoe/McGraw-Hill 383 Glencoe Algebra 2
Less
on
7-2
Complete each of the following.a. Graph each function by making a table of values.b. Determine consecutive values of x between which each real zero is located.c. Estimate the x-coordinates at which the relative maxima and minima occur.
1. f(x) � x3 � 3x2 � 1 2. f(x) � x3 � 3x � 1
zeros between �1 and 0, 0 and 1, zeros between �2 and �1, 0 and 1, and 2 and 3; rel. max. at x � 0, and 1 and 2; rel. max. at x � �1, rel. min. at x � 2 rel. min. at x � 1
3. f(x) � 2x3 � 9x2 �12x � 2 4. f(x) � 2x3 � 3x2 � 2
zero between �1 and 0; zero between �1 and 0; rel. max. at x � �2, rel. min. at x � 1, rel. max. at x � 0rel. min. at x � �1
5. f(x) � x4 � 2x2 � 2 6. f(x) � 0.5x4 � 4x2 � 4
zeros between �2 and �1, and zeros between �1 and �2, �2 and 1 and 2; rel. max. at x � 0, �3, 1 and 2, and 2 and 3; rel. max.at
x
f(x)
O
x f(x)
�3 8.5�2 �4�1 0.5
0 41 0.52 �43 8.5
x
f(x)
O
x f(x)
�3 61�2 6�1 �3
0 �21 �32 63 61
x
f(x)
O
x f(x)
�1 �30 21 12 63 29
x
f(x)
O
x f(x)
�3 �7�2 �2�1 �3
0 21 25
x
f(x)
O
x f(x)
�3 �17�2 �1�1 3
0 11 �12 33 19
x
f(x)
O
x f(x)
�2 �19�1 �3
0 11 �12 �33 14 17
© Glencoe/McGraw-Hill 384 Glencoe Algebra 2
Complete each of the following.a. Graph each function by making a table of values.b. Determine consecutive values of x between which each real zero is located.c. Estimate the x-coordinates at which the relative and relative minima occur.
1. f(x) � �x3 � 3x2 � 3 2. f(x) � x3 � 1.5x2 � 6x � 1
x
f(x)
O
8
4
–4
–8
2 4–2–4
x f(x)
�2 �1�1 4.5
0 11 �5.52 �93 �3.54 17
x
f(x)
O
x f(x)
�2 17�1 1
0 �31 �12 13 �34 �19
Practice (Average)
Graphing Polynomial Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
zeros between �1 zeros between �2 and 0, 1 and 2, and �1, 0 and 1,
and 2 and 3; rel. max. at x � 2, and 3 and 4; rel. max. at x � �1, rel. min. at x � 0 rel. min. at x � 2
3. f(x) � 0.75x4 � x3 � 3x2 � 4 4. f(x) � x4 � 4x3 � 6x2 � 4x � 3
zeros between �3 and �2, and zeros between �3 and �2, �2 and �1; rel. max. at x � 0, and 0 and 1; rel. min. at x � �1rel. min. at x � �2 and x � 1
PRICES For Exercises 5 and 6, use the following information.The Consumer Price Index (CPI) gives the relative price for a fixed set of goods and services. The CPI from September, 2000 to July, 2001 is shown in the graph.Source: U. S. Bureau of Labor Statistics
5. Describe the turning points of the graph.rel max. in Nov. and June; rel. min in Dec.
6. If the graph were modeled by a polynomial equation,what is the least degree the equation could have? 4
7. LABOR A town’s jobless rate can be modeled by (1, 3.3), (2, 4.9), (3, 5.3), (4, 6.4), (5, 4.5),(6, 5.6), (7, 2.5), (8, 2.7). How many turning points would the graph of a polynomialfunction through these points have? Describe them. 4: 2 rel. max. and 2 rel. min.
Months Since September, 2000
Co
nsu
mer
Pri
ce In
dex
20 4 61 3 5 7 8 9 1011
179178177176175174173
x f(x)
�3 12�2 �3�1 �4
0 �31 122 77
x f(x)
�3 10.75�2 �4�1 0.75
0 41 2.752 12
Reading to Learn MathematicsGraphing Polynomial Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
© Glencoe/McGraw-Hill 385 Glencoe Algebra 2
Less
on
7-2
Pre-Activity How can graphs of polynomial functions show trends in data?
Read the introduction to Lesson 7-2 at the top of page 353 in your textbook.
Three points on the graph shown in your textbook are (0, 14), (70, 3.78), and(100, 9). Give the real-world meaning of the coordinates of these points.Sample answer: In 1900, 14% of the U. S. population wasforeign born. In 1970, 3.78% of the population was foreignborn. In 2000, 9% of the population was foreign born.
Reading the Lesson
1. Suppose that f(x) is a third-degree polynomial function and that c and d are realnumbers, with d � c. Indicate whether each statement is true or false. (Remember thattrue means always true.)
a. If f(c) � 0 and f(d) � 0, there is exactly one real zero between c and d. false
b. If f(c) � f(d) 0, there are no real zeros between c and d. false
c. If f(c) � 0 and f(d) � 0, there is at least one real zero between c and d. true
2. Match each graph with its description.
a. third-degree polynomial with one relative maximum and one relative minimum;leading coefficient negative iii
b. fourth-degree polynomial with two relative minima and one relative maximum i
c. third-degree polynomial with one relative maximum and one relative minimum;leading coefficient positive iv
d. fourth-degree polynomial with two relative maxima and one relative minimum ii
i. ii. iii. iv.
Helping You Remember
3. The origins of words can help you to remember their meaning and to distinguishbetween similar words. Look up maximum and minimum in a dictionary and describetheir origins (original language and meaning). Sample answer: Maximum comesfrom the Latin word maximus, meaning greatest. Minimum comes fromthe Latin word minimus, meaning least.
x
f(x)
Ox
f(x)
Ox
f(x)
Ox
f(x)
O
© Glencoe/McGraw-Hill 386 Glencoe Algebra 2
Golden RectanglesUse a straightedge, a compass, and the instructions below to construct a golden rectangle.
1. Construct square ABCD with sides of 2 centimeters.
2. Construct the midpoint of A�B�. Call the midpoint M.
3. Using M as the center, set your compass opening at MC. Construct an arc with center M that intersects A�B�. Call the point of intersection P.
4. Construct a line through P that is perpendicular to A�B�.
5. Extend D�C� so that it intersects the perpendicular. Call the intersection point Q.APQD is a golden rectangle. Check this
conclusion by finding the value of �QAP
P�.
A figure consisting of similar golden rectangles is shown below. Use a compass and the instructions below to draw quarter-circle arcs that form a spiral like that found in the shell of a chambered nautilus.
6. Using A as a center, draw an arc that passes through B and C.
7. Using D as a center, draw an arc that passes through C and E.
8. Using F as a center, draw an arc that passes through E and G.
9. Continue drawing arcs,using H, K, and M as the centers.
C
BA G
HJ
D E
K
M
L F
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
Study Guide and InterventionSolving Equations Using Quadratic Techniques
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
© Glencoe/McGraw-Hill 387 Glencoe Algebra 2
Less
on
7-3
Quadratic Form Certain polynomial expressions in x can be written in the quadraticform au2 � bu � c for any numbers a, b, and c, a 0, where u is an expression in x.
Write each polynomial in quadratic form, if possible.
a. 3a6 � 9a3 � 12Let u � a3.3a6 � 9a3 � 12 � 3(a3)2 � 9(a3) � 12
b. 101b � 49�b� � 42Let u � �b�.101b � 49�b� � 42 � 101(�b�)2
� 49(�b�) � 42
c. 24a5 � 12a3 � 18This expression cannot be written in quadratic form, since a5 (a3)2.
Write each polynomial in quadratic form, if possible.
1. x4 � 6x2 � 8 2. 4p4 � 6p2 � 8
(x2)2 � 6(x2) � 8 4(p2)2 � 6(p2) � 8
3. x8 � 2x4 � 1 4. x�18
�� 2x�
116�
� 1
(x4)2 � 2(x4) � 1 �x�116��2
� 2�x�116�� � 1
5. 6x4 � 3x3 � 18 6. 12x4 � 10x2 � 4
not possible 12(x2)2 � 10(x2) � 4
7. 24x8 � x4 � 4 8. 18x6 � 2x3 � 12
24(x4)2 � x4 � 4 18(x3)2 � 2(x3) � 12
9. 100x4 � 9x2 � 15 10. 25x8 � 36x6 � 49
100(x2)2 � 9(x2) � 15 not possible
11. 48x6 � 32x3 � 20 12. 63x8 � 5x4 � 29
48(x3)2 � 32(x3) � 20 63(x4)2 � 5(x4) � 29
13. 32x10 � 14x5 � 143 14. 50x3 � 15x�x� � 18
32(x5)2 � 14(x5) � 143 50�x�32
��2� 15�x�
32
�� � 18
15. 60x6 � 7x3 � 3 16. 10x10 � 7x5 � 7
60(x3)2 � 7(x3) � 3 10(x5)2 � 7(x5) � 7
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 388 Glencoe Algebra 2
Solve Equations Using Quadratic Form If a polynomial expression can be writtenin quadratic form, then you can use what you know about solving quadratic equations tosolve the related polynomial equation.
Solve x4 � 40x2 � 144 � 0.x4 � 40x2 � 144 � 0 Original equation
(x2)2 � 40(x2) � 144 � 0 Write the expression on the left in quadratic form.
(x2 � 4)(x2 � 36) � 0 Factor.x2 � 4 � 0 or x2 � 36 � 0 Zero Product Property
(x � 2)(x � 2) � 0 or (x � 6)(x � 6) � 0 Factor.
x � 2 � 0 or x � 2 � 0 or x � 6 � 0 or x � 6 � 0 Zero Product Property
x � 2 or x � �2 or x � 6 or x � �6 Simplify.
The solutions are 2 and 6.
Solve 2x � �x� � 15 � 0.2x � �x� � 15 � 0 Original equation
2(�x�)2 � �x� � 15 � 0 Write the expression on the left in quadratic form.
(2�x� �5)(�x� � 3) � 0 Factor.
2�x� � 5 � 0 or �x� � 3 � 0 Zero Product Property
�x� � or �x� � �3 Simplify.
Since the principal square root of a number cannot be negative, �x� � �3 has no solution.
The solution is or 6 .
Solve each equation.
1. x4 � 49 2. x4 � 6x2 � �8 3. x4 � 3x2 � 54
��7�, �i �7� �2, ��2� �3, �i �6�
4. 3t6 � 48t2 � 0 5. m6 � 16m3 � 64 � 0 6. y4 � 5y2 � 4 � 0
0, �2, �2i 2, �1 � i �3� �1, �2
7. x4 � 29x2 � 100 � 0 8. 4x4 � 73x2 � 144 � 0 9. � � 12 � 0
�5, �2 �4, � ,
10. x � 5�x� � 6 � 0 11. x � 10�x� � 21 � 0 12. x�23
�� 5x�
13
�� 6 � 0
4, 9 9, 49 27, 8
1�
1�
3�
7�x
1�x2
1�4
25�4
5�2
Study Guide and Intervention (continued)
Solving Equations Using Quadratic Techniques
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
Example 1Example 1
Example 2Example 2
ExercisesExercises
Skills PracticeSolving Equations Using Quadratic Techniques
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
© Glencoe/McGraw-Hill 389 Glencoe Algebra 2
Less
on
7-3
Write each expression in quadratic form, if possible.
1. 5x4 � 2x2 � 8 5(x2)2 � 2(x2) � 8 2. 3y8 � 4y2 � 3 not possible
3. 100a6 � a3 100(a3)2 � a3 4. x8 � 4x4 � 9 (x4)2 � 4(x4) � 9
5. 12x4 � 7x2 12(x2)2 � 7(x2) 6. 6b5 � 3b3 � 1 not possible
7. 15v6 � 8v3 � 9 15(v3)2 � 8(v3) � 9 8. a9 � 5a5 � 7a a[(a4)2 � 5(a4) � 7]
Solve each equation.
9. a3 � 9a2 � 14a � 0 0, 7, 2 10. x3 � 3x2 0, 3
11. t4 � 3t3 � 40t2 � 0 0, �5, 8 12. b3 � 8b2 � 16b � 0 0, 4
13. m4 � 4 ��2�, �2�, �i�2�, i�2� 14. w3 � 6w � 0 0, �6�, ��6�
15. m4 � 18m2 � �81 �3, 3 16. x5 � 81x � 0 0, �3, 3, �3i, 3i
17. h4 � 10h2 � �9 �1, 1, �3, 3 18. a4 � 9a2 � 20 � 0 �2, 2, �5�, ��5�
19. y4 � 7y2 � 12 � 0 20. v4 � 12v2 � 35 � 02, �2, �3�, ��3� �5�, ��5�, �7�, ��7�
21. x5 � 7x3 � 6x � 0 22. c�23
�� 7c�
13
�� 12 � 0
0, �1, 1, �6�, ��6� �64, �27
23. z � 5�z� � �6 4, 9 24. x � 30�x� � 200 � 0 100, 400
© Glencoe/McGraw-Hill 390 Glencoe Algebra 2
Write each expression in quadratic form, if possible.
1. 10b4 � 3b2 � 11 2. �5x8 � x2 � 6 3. 28d6 � 25d3
10(b2)2 � 3(b2) � 11 not possible 28(d3)2 � 25(d3)
4. 4s8 � 4s4 � 7 5. 500x4 � x2 6. 8b5 � 8b3 � 1
4(s4)2 � 4(s4) � 7 500(x2)2 � x2 not possible
7. 32w5 � 56w3 � 8w 8. e�23
�� 7e�
13
�� 10 9. x
�15
�� 29x
�110�
� 2
8w[4(w2)2 � 7(w2) � 1] (e�13
�)2� 7(e�
13
�) � 10 (x�110�)2
� 29(x�110�) � 2
Solve each equation.
10. y4 � 7y3 � 18y2 � 0 �2, 0, 9 11. s5 � 4s4 � 32s3 � 0 �8, 0, 4
12. m4 � 625 � 0 �5, 5, �5i, 5i 13. n4 � 49n2 � 0 0, �7, 7
14. x4 � 50x2 � 49 � 0 �1, 1, �7, 7 15. t4 � 21t2 � 80 � 0 �4, 4, �5�, ��5�
16. 4r6 � 9r4 � 0 0, �32
�, ��32
� 17. x4 � 24 � �2x2 �2, 2, �i�6�, i�6�
18. d4 � 16d2 � 48 �2, 2, �2�3�, 2�3� 19. t3 � 343 � 0 7, ,
20. x�12
�� 5x
�14
�� 6 � 0 16, 81 21. x
�43
�� 29x
�23
�� 100 � 0 8, 125
22. y3 � 28y�32
�� 27 � 0 1, 9 23. n � 10�n� � 25 � 0 25
24. w � 12�w� � 27 � 0 9, 81 25. x � 2�x� � 80 � 0 100
26. PHYSICS A proton in a magnetic field follows a path on a coordinate grid modeled bythe function f(x) � x4 � 2x2 � 15. What are the x-coordinates of the points on the gridwhere the proton crosses the x-axis? ��5�, �5�
27. SURVEYING Vista county is setting aside a large parcel of land to preserve it as openspace. The county has hired Meghan’s surveying firm to survey the parcel, which is inthe shape of a right triangle. The longer leg of the triangle measures 5 miles less thanthe square of the shorter leg, and the hypotenuse of the triangle measures 13 miles lessthan twice the square of the shorter leg. The length of each boundary is a whole number.Find the length of each boundary. 3 mi, 4 mi, 5 mi
�7 � 7i�3���
�7 � 7i�3���
Practice (Average)
Solving Equations Using Quadratic Techniques
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
Reading to Learn MathematicsSolving Equations Using Quadratic Techniques
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
© Glencoe/McGraw-Hill 391 Glencoe Algebra 2
Less
on
7-3
Pre-Activity How can solving polynomial equations help you to find dimensions?
Read the introduction to Lesson 7-3 at the top of page 360 in your textbook.
Explain how the formula given for the volume of the box can be obtainedfrom the dimensions shown in the figure.
Sample answer: The volume of a rectangular box is given by the formula V � �wh. Substitute 50 � 2x for �, 32 � 2x for w, and x for h to get V(x) � (50 � 2x)(32 � 2x)(x) � 4x3 � 164x2 � 1600x.
Reading the Lesson
1. Which of the following expressions can be written in quadratic form? b, c, d, f, g, h, i
a. x3 � 6x2 � 9 b. x4 � 7x2 � 6 c. m6 � 4m3 � 4
d. y � 2y�12
�� 15 e. x5 � x3 � 1 f. r4 � 6 � r8
g. p�14
�� 8p
�12
�� 12 h. r
�13
�� 2r
�16
�� 3 i. 5�z� � 2z � 3
2. Match each expression from the list on the left with its factorization from the list on the right.
a. x4 � 3x2 � 40 vi i. (x3 � 3)(x3 � 3)
b. x4 � 10x2 � 25 v ii. (�x� � 3)(�x� � 3)
c. x6 � 9 i iii. (�x� � 3)2
d. x � 9 ii iv. (x2 � 1)(x4 � x2 � 1)
e. x6 � 1 iv v. (x2 � 5)2
f. x � 6�x� � 9 iii vi. (x2 � 5)(x2 � 8)
Helping You Remember
3. What is an easy way to tell whether a trinomial in one variable containing one constantterm can be written in quadratic form?
Sample answer: Look at the two terms that are not constants andcompare the exponents on the variable. If one of the exponents is twicethe other, the trinomial can be written in quadratic form.
© Glencoe/McGraw-Hill 392 Glencoe Algebra 2
Odd and Even Polynomial Functions
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
Functions whose graphs are symmetric withrespect to the origin are called odd functions.If f(�x) � �f(x) for all x in the domain of f(x),then f (x) is odd.
Functions whose graphs are symmetric withrespect to the y-axis are called even functions.If f (�x) � f(x) for all x in the domain of f(x),then f (x) is even.
x
f(x)
O 1 2–2 –1
6
4
2f(x) � 1–4x4 � 4
x
f(x)
O 1 2–2 –1
4
2
–2
–4
f(x) � 1–2x3
ExampleExample Determine whether f(x) � x3 � 3x is odd, even, or neither.
f(x) � x3 � 3xf(�x) � (�x)3 � 3(�x) Replace x with �x.
� �x3 � 3x Simplify.
� �(x3 � 3x) Factor out �1.
� �f (x) Substutute.
Therefore, f (x) is odd.
The graph at the right verifies that f (x) is odd.The graph of the function is symmetric with respect to the origin.
Determine whether each function is odd, even, or neither by graphing or by applying the rules for odd and even functions.
1. f (x) � 4x2 2. f (x) � �7x4
3. f (x) � x7 4. f (x) � x3 � x2
5. f (x) � 3x3 � 1 6. f (x) � x8 � x5 � 6
7. f (x) � �8x5 � 2x3 � 6x 8. f (x) � x4 � 3x3 � 2x2 � 6x � 1
9. f (x) � x4 � 3x2 � 11 10. f (x) � x7 � 6x5 � 2x3 � x
11. Complete the following definitions: A polynomial function is odd if and only
if all the terms are of degrees. A polynomial function is even
if and only if all the terms are of degrees.
x
f(x)
O 1 2–2 –1
4
2
–2
–4
f(x) � x3 � 3x
Study Guide and InterventionThe Remainder and Factor Theorems
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
© Glencoe/McGraw-Hill 393 Glencoe Algebra 2
Less
on
7-4
Synthetic Substitution
Remainder The remainder, when you divide the polynomial f(x ) by (x � a), is the constant f(a).Theorem f(x) � q(x ) � (x � a) � f(a), where q(x) is a polynomial with degree one less than the degree of f(x).
If f(x) � 3x4 � 2x3 � 5x2 � x � 2, find f(�2).Example 1Example 1
Example 2Example 2
Method 1 Synthetic SubstitutionBy the Remainder Theorem, f(�2) shouldbe the remainder when you divide thepolynomial by x � 2.
�2 3 2 �5 1 �2�6 8 �6 10
3 �4 3 �5 8The remainder is 8, so f(�2) � 8.
Method 2 Direct SubstitutionReplace x with �2.
f(x) � 3x4 � 2x3 � 5x2 � x � 2f(�2) � 3(�2)4 � 2(�2)3 � 5(�2)2 � (�2) � 2
� 48 � 16 � 20 � 2 � 2 or 8So f(�2) � 8.
If f(x) � 5x3 � 2x � 1, find f(3).Again, by the Remainder Theorem, f(3) should be the remainder when you divide thepolynomial by x � 3.
3 5 0 2 �115 45 141
5 15 47 140The remainder is 140, so f(3) � 140.
Use synthetic substitution to find f(�5) and f � � for each function.
1. f(x) � �3x2 � 5x � 1 �101; 2. f(x) � 4x2 � 6x � 7 63; �3
3. f(x) � �x3 � 3x2 � 5 195; � 4. f(x) � x4 � 11x2 � 1 899;
Use synthetic substitution to find f(4) and f(�3) for each function.
5. f(x) � 2x3 � x2 � 5x � 3 6. f(x) � 3x3 � 4x � 2127; �27 178; �67
7. f(x) � 5x3 � 4x2 � 2 8. f(x) � 2x4 � 4x3 � 3x2 � x � 6258; �169 302; 288
9. f(x) � 5x4 � 3x3 � 4x2 � 2x � 4 10. f(x) � 3x4 � 2x3 � x2 � 2x � 51404; 298 627; 277
11. f(x) � 2x4 � 4x3 � x2 � 6x � 3 12. f(x) � 4x4 � 4x3 � 3x2 � 2x � 3219; 282 805; 462
29�
35�
3�
1�2
ExercisesExercises
© Glencoe/McGraw-Hill 394 Glencoe Algebra 2
Factors of Polynomials The Factor Theorem can help you find all the factors of apolynomial.
Factor Theorem The binomial x � a is a factor of the polynomial f(x) if and only if f(a) � 0.
Show that x � 5 is a factor of x3 � 2x2 � 13x � 10. Then find theremaining factors of the polynomial.By the Factor Theorem, the binomial x � 5 is a factor of the polynomial if �5 is a zero of thepolynomial function. To check this, use synthetic substitution.
�5 1 2 �13 10�5 15 �10
1 �3 2 0
Since the remainder is 0, x � 5 is a factor of the polynomial. The polynomial x3 � 2x2 � 13x � 10 can be factored as (x � 5)(x2 � 3x � 2). The depressed polynomial x2 � 3x � 2 can be factored as (x � 2)(x � 1).
So x3 � 2x2 � 13x � 10 � (x � 5)(x � 2)(x � 1).
Given a polynomial and one of its factors, find the remaining factors of thepolynomial. Some factors may not be binomials.
1. x3 � x2 � 10x � 8; x � 2 2. x3 � 4x2 � 11x � 30; x � 3(x � 4)(x � 1) (x � 5)(x � 2)
3. x3 � 15x2 � 71x � 105; x � 7 4. x3 � 7x2 � 26x � 72; x � 4(x � 3)(x � 5) (x � 2)(x � 9)
5. 2x3 � x2 � 7x � 6; x � 1 6. 3x3 � x2 � 62x � 40; x � 4(2x � 3)(x � 2) (3x � 2)(x � 5)
7. 12x3 � 71x2 � 57x � 10; x � 5 8. 14x3 � x2 � 24x � 9; x � 1(4x � 1)(3x � 2) (7x � 3)(2x � 3)
9. x3 � x � 10; x � 2 10. 2x3 � 11x2 � 19x � 28; x � 4(x2 � 2x � 5) (2x2 � 3x � 7)
11. 3x3 � 13x2 � 34x � 24; x � 6 12. x4 � x3 � 11x2 � 9x � 18; x � 1(3x2 � 5x � 4) (x � 2)(x � 3)(x � 3)
Study Guide and Intervention (continued)
The Remainder and Factor Theorems
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
ExampleExample
ExercisesExercises
Skills PracticeThe Remainder and Factor Theorems
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
© Glencoe/McGraw-Hill 395 Glencoe Algebra 2
Less
on
7-4
Use synthetic substitution to find f(2) and f(�1) for each function.
1. f(x) � x2 � 6x � 5 21, 0 2. f(x) � x2 � x � 1 3, 3
3. f(x) � x2 � 2x � 2 �2, 1 4. f(x) � x3 � 2x2 � 5 21, 6
5. f(x) � x3 � x2 � 2x � 3 3, 3 6. f(x) � x3 � 6x2 � x � 4 30, 0
7. f(x) � x3 � 3x2 � x � 2 �4, �7 8. f(x) � x3 � 5x2 � x � 6 �8, 1
9. f(x) � x4 � 2x2 � 9 15, �6 10. f(x) � x4 � 3x3 � 2x2 � 2x � 6 2, 14
11. f(x) � x5 � 7x3 � 4x � 10 12. f(x) � x6 � 2x5 � x4 � x3 � 9x2 � 20�22, 20 �32, �26
Given a polynomial and one of its factors, find the remaining factors of thepolynomial. Some factors may not be binomials.
13. x3 � 2x2 � x � 2; x � 1 14. x3 � x2 � 5x � 3; x � 1
x � 1, x � 2 x � 1, x � 3
15. x3 � 3x2 � 4x � 12; x � 3 16. x3 � 6x2 � 11x � 6; x � 3
x � 2, x � 2 x � 1, x � 2
17. x3 � 2x2 � 33x � 90; x � 5 18. x3 � 6x2 � 32; x � 4
x � 3, x � 6 x � 4, x � 2
19. x3 � x2 � 10x � 8; x � 2 20. x3 � 19x � 30; x � 2
x � 1, x � 4 x � 5, x � 3
21. 2x3 � x2 � 2x � 1; x � 1 22. 2x3 � x2 � 5x � 2; x � 2
2x � 1, x � 1 x � 1, 2x � 1
23. 3x3 � 4x2 � 5x � 2; 3x � 1 24. 3x3 � x2 � x � 2; 3x � 2
x � 1, x � 2 x2 � x � 1
© Glencoe/McGraw-Hill 396 Glencoe Algebra 2
Use synthetic substitution to find f(�3) and f(4) for each function.
1. f(x) � x2 � 2x � 3 6, 27 2. f(x) � x2 � 5x � 10 34, 6
3. f(x) � x2 � 5x � 4 20, �8 4. f(x) � x3 � x2 � 2x � 3 �27, 43
5. f(x) � x3 � 2x2 � 5 �4, 101 6. f(x) � x3 � 6x2 � 2x �87, �24
7. f(x) � x3 � 2x2 � 2x � 8 �31, 32 8. f(x) � x3 � x2 � 4x � 4 �52, 60
9. f(x) � x3 � 3x2 � 2x � 50 �56, 70 10. f(x) � x4 � x3 � 3x2 � x � 12 42, 280
11. f(x) � x4 � 2x2 � x � 7 73, 227 12. f(x) � 2x4 � 3x3 � 4x2 � 2x � 1 286, 377
13. f(x) � 2x4 � x3 � 2x2 � 26 181, 454 14. f(x) � 3x4 � 4x3 � 3x2 � 5x � 3 390, 537
15. f(x) � x5 � 7x3 � 4x � 10 16. f(x) � x6 � 2x5 � x4 � x3 � 9x2 � 20�430, 1446 74, 5828
Given a polynomial and one of its factors, find the remaining factors of thepolynomial. Some factors may not be binomials.
17. x3 � 3x2 � 6x � 8; x � 2 18. x3 � 7x2 � 7x � 15; x � 1
x � 1, x � 4 x � 3, x � 5
19. x3 � 9x2 � 27x � 27; x � 3 20. x3 � x2 � 8x � 12; x � 3
x � 3, x � 3 x � 2, x � 2
21. x3 � 5x2 � 2x � 24; x � 2 22. x3 � x2 � 14x � 24; x � 4
x � 3, x � 4 x � 3, x � 2
23. 3x3 � 4x2 � 17x � 6; x � 2 24. 4x3 � 12x2 � x � 3; x � 3
x � 3, 3x � 1 2x � 1, 2x � 1
25. 18x3 � 9x2 � 2x � 1; 2x � 1 26. 6x3 � 5x2 � 3x � 2; 3x � 2
3x � 1, 3x � 1 2x � 1, x � 1
27. x5 � x4 � 5x3 � 5x2 � 4x � 4; x � 1 28. x5 � 2x4 � 4x3 � 8x2 � 5x � 10; x � 2
x � 1, x � 1, x � 2, x � 2 x � 1, x � 1, x2 � 5
29. POPULATION The projected population in thousands for a city over the next severalyears can be estimated by the function P(x) � x3 � 2x2 � 8x � 520, where x is thenumber of years since 2000. Use synthetic substitution to estimate the population for 2005. 655,000
30. VOLUME The volume of water in a rectangular swimming pool can be modeled by thepolynomial 2x3 � 9x2 � 7x � 6. If the depth of the pool is given by the polynomial 2x � 1, what polynomials express the length and width of the pool? x � 3 and x � 2
Practice (Average)
The Remainder and Factor Theorems
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
Reading to Learn MathematicsThe Remainder and Factor Theorems
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
© Glencoe/McGraw-Hill 397 Glencoe Algebra 2
Less
on
7-4
Pre-Activity How can you use the Remainder Theorem to evaluate polynomials?
Read the introduction to Lesson 7-4 at the top of page 365 in your textbook.
Show how you would use the model in the introduction to estimate thenumber of international travelers (in millions) to the United States in theyear 2000. (Show how you would substitute numbers, but do not actuallycalculate the result.)Sample answer: 0.02(14)3 � 0.6(14)2 � 6(14) � 25.9
Reading the Lesson
1. Consider the following synthetic division.1 3 2 �6 4
3 5 �13 5 �1 3
a. Using the division symbol �, write the division problem that is represented by thissynthetic division. (Do not include the answer.) (3x3 � 2x2 � 6x � 4) � (x � 1)
b. Identify each of the following for this division.
dividend divisor
quotient remainder
c. If f(x) � 3x3 � 2x2 � 6x � 4, what is f(1)? 3
2. Consider the following synthetic division.�3 1 0 0 27
�3 9 �271 �3 9 0
a. This division shows that is a factor of .
b. The division shows that is a zero of the polynomial function
f(x) � .
c. The division shows that the point is on the graph of the polynomial
function f(x) � .
Helping You Remember
3. Think of a mnemonic for remembering the sentence, “Dividend equals quotient timesdivisor plus remainder.”Sample answer: Definitely every quiet teacher deserves proper rewards.
x3 � 27(�3, 0)
x3 � 27�3
x3 � 27x � 3
33x3 � 5x � 1
x � 13x3 � 2x2 � 6x � 4
© Glencoe/McGraw-Hill 398 Glencoe Algebra 2
Using Maximum ValuesMany times maximum solutions are needed for different situations. For instance, what is the area of the largest rectangular field that can be enclosed with 2000 feet of fencing?
Let x and y denote the length and width of the field, respectively.
Perimeter: 2x � 2y � 2000 → y � 1000 � xArea: A � xy � x(1000 � x) � �x2 � 1000x
This problem is equivalent to finding the highest point on the graph of A(x) � �x2 � 1000x shown on the right.
Complete the square for �x2 � 1000x.
A � �(x2 � 1000x � 5002) � 5002
� �(x � 500)2 � 5002
Because the term �(x � 500)2 is either negative or 0, the greatest value of Ais 5002. The maximum area enclosed is 5002 or 250,000 square feet.
Solve each problem.
1. Find the area of the largest rectangular garden that can be enclosed by 300 feet of fence.
2. A farmer will make a rectangular pen with 100 feet of fence using part of his barn for one side of the pen. What is the largest area he can enclose?
3. An area along a straight stone wall is to be fenced. There are 600 meters of fencing available. What is the greatest rectangular area that can be enclosed?
A
xO 1000
x
y
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
Study Guide and InterventionRoots and Zeros
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
© Glencoe/McGraw-Hill 399 Glencoe Algebra 2
Less
on
7-5
Types of Roots The following statements are equivalent for any polynomial function f(x).• c is a zero of the polynomial function f(x).• (x � c) is a factor of the polynomial f(x).• c is a root or solution of the polynomial equation f(x) � 0.If c is real, then (c, 0) is an intercept of the graph of f(x).
Fundamental Every polynomial equation with degree greater than zero has at least one root in the setTheorem of Algebra of complex numbers.
Corollary to the A polynomial equation of the form P (x) � 0 of degree n with complex coefficients hasFundamental exactly n roots in the set of complex numbers.Theorem of Algebras
If P (x) is a polynomial with real coefficients whose terms are arranged in descendingpowers of the variable,
Descartes’ Rule• the number of positive real zeros of y � P (x) is the same as the number of changes in
of Signssign of the coefficients of the terms, or is less than this by an even number, and
• the number of negative real zeros of y � P (x) is the same as the number of changes in sign of the coefficients of the terms of P (�x), or is less than this number by an evennumber.
Solve the equation 6x3 � 3x � 0 and state thenumber and type of roots.
6x3 � 3x � 03x(2x2 � 1) � 0Use the Zero Product Property.3x � 0 or 2x2 � 1 � 0x � 0 or 2x2 � �1
x �
The equation has one real root, 0,
and two imaginary roots, .i�2��2
i�2��2
State the number of positivereal zeros, negative real zeros, and imaginaryzeros for p(x) � 4x4 � 3x3 � x2 � 2x � 5.Since p(x) has degree 4, it has 4 zeros.Use Descartes’ Rule of Signs to determine thenumber and type of real zeros. Since there are threesign changes, there are 3 or 1 positive real zeros.Find p(�x) and count the number of changes insign for its coefficients.p(�x) � 4(�x)4 � 3(�x)3 � (�x)2 � 2(�x) � 5
� 4x4 � 3x3 � x2 � 2x � 5Since there is one sign change, there is exactly 1negative real zero.
Example 1Example 1 Example 2Example 2
ExercisesExercises
Solve each equation and state the number and type of roots.
1. x2 � 4x � 21� 0 2. 2x3 � 50x � 0 3. 12x3 � 100x � 0
3, �7; 2 real 0, �5; 3 real 0, � ; 1 real, 2imaginary
State the number of positive real zeros, negative real zeros, and imaginary zerosfor each function.
4. f(x) � 3x3 � x2 � 8x � 12 1; 2 or 0; 0 or 2
5. f(x) � 2x4 � x3 � 3x � 7 2 or 0; 0; 2 or 4
5i �3��
© Glencoe/McGraw-Hill 400 Glencoe Algebra 2
Find Zeros
Complex Conjugate Suppose a and b are real numbers with b 0. If a � bi is a zero of a polynomial Theorem function with real coefficients, then a � bi is also a zero of the function.
Find all of the zeros of f(x) � x4 � 15x2 � 38x � 60.Since f(x) has degree 4, the function has 4 zeros.f(x) � x4 � 15x2 � 38x � 60 f(�x) � x4 � 15x2 � 38x � 60Since there are 3 sign changes for the coefficients of f(x), the function has 3 or 1 positive realzeros. Since there is 1 sign change for the coefficients of f(�x), the function has 1 negativereal zero. Use synthetic substitution to test some possible zeros.
2 1 0 �15 38 �602 4 �22 32
1 2 �11 16 �28
3 1 0 �15 38 �603 9 �18 60
1 3 �6 20 0So 3 is a zero of the polynomial function. Now try synthetic substitution again to find a zeroof the depressed polynomial.
�2 1 3 �6 20�2 �2 16
1 1 �8 36
�4 1 3 �6 20�4 4 8
1 �1 �2 28
�5 1 3 �6 20�5 10 �20
1 �2 4 0
So � 5 is another zero. Use the Quadratic Formula on the depressed polynomial x2 � 2x � 4 to find the other 2 zeros, 1 i�3�.The function has two real zeros at 3 and �5 and two imaginary zeros at 1 i�3�.
Find all of the zeros of each function.
1. f(x) � x3 � x2 � 9x � 9 �1, �3i 2. f(x) � x3 � 3x2 � 4x � 12 3, �2i
3. p(a) � a3 � 10a2 � 34a � 40 4, 3 � i 4. p(x) � x3 � 5x2 � 11x � 15 3, 1 � 2i
5. f(x) � x3 � 6x � 20 6. f(x) � x4 � 3x3 � 21x2 � 75x � 100�2, 1 � 3i �1, 4, �5i
Study Guide and Intervention (continued)
Roots and Zeros
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
ExampleExample
ExercisesExercises
Skills PracticeRoots and Zeros
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
© Glencoe/McGraw-Hill 401 Glencoe Algebra 2
Less
on
7-5
Solve each equation. State the number and type of roots.
1. 5x � 12 � 0 2. x2 � 4x � 40 � 0
��152�; 1 real 2 � 6i; 2 imaginary
3. x5 � 4x3 � 0 4. x4 � 625 � 0
0, 0, 0, 2i, �2i; 3 real, 2 imaginary 5i, 5i, �5i, �5i; 4 imaginary
5. 4x2 � 4x � 1 � 0 6. x5 � 81x � 0
; 2 real 0, �3, 3, �3i, 3i; 3 real, 2 imaginary
State the possible number of positive real zeros, negative real zeros, andimaginary zeros of each function.
7. g(x) � 3x3 � 4x2 � 17x � 6 8. h(x) � 4x3 � 12x2 � x � 3
2 or 0; 1; 2 or 0 2 or 0; 1; 2 or 0
9. f(x) � x3 � 8x2 � 2x � 4 10. p(x) � x3 � x2 � 4x � 6
3 or 1; 0; 2 or 0 3 or 1; 0; 2 or 0
11. q(x) � x4 � 7x2 � 3x � 9 12. f(x) � x4 � x3 � 5x2 � 6x � 1
1; 1; 2 2 or 0; 2 or 0; 4 or 2 or 0
Find all the zeros of each function.
13. h(x) � x3 � 5x2 � 5x � 3 14. g(x) � x3 � 6x2 � 13x � 10
3, 1 � �2�, 1 � �2� 2, 2 � i, 2 � i
15. h(x) � x3 � 4x2 � x � 6 16. q(x) � x3 � 3x2 � 6x � 8
1, �2, �3 2, �1, �4
17. g(x) � x4 � 3x3 � 5x2 � 3x � 4 18. f(x) � x4 � 21x2 � 80
�1, �1, 1, 4 �4, 4, ��5�, �5�
Write a polynomial function of least degree with integral coefficients that has thegiven zeros.
19. �3, �5, 1 20. 3if(x) � x3 � 7x2 � 7x � 15 f(x) � x2 � 9
21. �5 � i 22. �1, �3�, ��3�f(x) � x2 � 10x � 26 f(x) � x3 � x2 � 3x � 3
23. i, 5i 24. �1, 1, i�6�f(x) � x4 � 26x2 � 25 f(x) � x4 � 5x2 � 6
1 � �2��
© Glencoe/McGraw-Hill 402 Glencoe Algebra 2
Solve each equation. State the number and type of roots.
1. �9x � 15 � 0 2. x4 � 5x2 � 4 � 0
��53
�; 1 real �1, 1, �2, 2; 4 real
3. x5 � 81x 4. x3 � x2 � 3x � 3 � 0
0, �3, 3, �3i, 3i; 3 real, 2 imaginary �1, ��3�, �3�; 3 real
5. x3 � 6x � 20 � 0 6. x4 � x3 � x2 � x � 2 � 0
�2, 1 � 3i; 1 real, 2 imaginary 2, �1, �i, i; 2 real, 2 imaginary
State the possible number of positive real zeros, negative real zeros, andimaginary zeros of each function.
7. f(x) � 4x3 � 2x2 � x � 3 8. p(x) � 2x4 � 2x3 � 2x2 � x � 1
2 or 0; 1; 2 or 0 3 or 1; 1; 2 or 0
9. q(x) � 3x4 � x3 � 3x2 � 7x � 5 10. h(x) � 7x4 � 3x3 � 2x2 � x � 1
2 or 0; 2 or 0; 4, 2, or 0 2 or 0; 2 or 0; 4, 2, or 0
Find all the zeros of each function.
11. h(x) � 2x3 � 3x2 � 65x � 84 12. p(x) � x3 � 3x2 � 9x � 7
�7, �32
�, 4 1, 1 � i�6�, 1 � i�6�
13. h(x) � x3 � 7x2 � 17x � 15 14. q(x) � x4 � 50x2 � 49
3, 2 � i, 2 � i �i, i, �7i, 7i
15. g(x) � x4 � 4x3 � 3x2 � 14x � 8 16. f(x) � x4 � 6x3 � 6x2 � 24x � 40
�1, �1, 2, �4 �2, 2, 3 � i, 3 � i
Write a polynomial function of least degree with integral coefficients that has thegiven zeros.
17. �5, 3i 18. �2, 3 � if(x) � x3 � 5x2 � 9x � 45 f(x) � x3 � 4x2 � 2x � 20
19. �1, 4, 3i 20. 2, 5, 1 � if(x) � x4 � 3x3 � 5x2 � 27x � 36 f(x) � x4 � 9x3 � 26x2 � 34x � 20
21. CRAFTS Stephan has a set of plans to build a wooden box. He wants to reduce thevolume of the box to 105 cubic inches. He would like to reduce the length of eachdimension in the plan by the same amount. The plans call for the box to be 10 inches by8 inches by 6 inches. Write and solve a polynomial equation to find out how muchStephen should take from each dimension. (10 � x)(8 � x)(6 � x) � 105; 3 in.
Practice (Average)
Roots and Zeros
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
Reading to Learn MathematicsRoots and Zeros
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
© Glencoe/McGraw-Hill 403 Glencoe Algebra 2
Less
on
7-5
Pre-Activity How can the roots of an equation be used in pharmacology?
Read the introduction to Lesson 7-5 at the top of page 371 in your textbook.
Using the model given in the introduction, write a polynomial equationwith 0 on one side that can be solved to find the time or times at whichthere is 100 milligrams of medication in a patient’s bloodstream.0.5t4 � 3.5t3 � 100t2 � 350t � 100 � 0
Reading the Lesson
1. Indicate whether each statement is true or false.
a. Every polynomial equation of degree greater than one has at least one root in the setof real numbers. false
b. If c is a root of the polynomial equation f(x) � 0, then (x � c) is a factor of thepolynomial f(x). true
c. If (x � c) is a factor of the polynomial f(x), then c is a zero of the polynomial function f. false
d. A polynomial function f of degree n has exactly (n � 1) complex zeros. false
2. Let f(x) � x6 � 2x5 � 3x4 � 4x3 � 5x2 � 6x � 7.
a. What are the possible numbers of positive real zeros of f ? 5, 3, or 1b. Write f(�x) in simplified form (with no parentheses).
x6 � 2x5 � 3x4 � 4x3 � 5x2 � 6x � 7What are the possible numbers of negative real zeros of f ? 1
c. Complete the following chart to show the possible combinations of positive real zeros,negative real zeros, and imaginary zeros of the polynomial function f.
Number of Number of Number of Total Number Positive Real Zeros Negative Real Zeros Imaginary Zeros of Zeros
5 1 0 6
3 1 2 6
1 1 4 6
Helping You Remember
3. It is easier to remember mathematical concepts and results if you relate them to eachother. How can the Complex Conjugates Theorem help you remember the part ofDescartes’ Rule of Signs that says, “or is less than this number by an even number.”Sample answer: For a polynomial function in which the polynomial hasreal coefficients, imaginary zeros come in conjugate pairs. Therefore,there must be an even number of imaginary zeros. For each pair ofimaginary zeros, the number of positive or negative zeros decreases by
© Glencoe/McGraw-Hill 404 Glencoe Algebra 2
The Bisection Method for Approximating Real ZerosThe bisection method can be used to approximate zeros of polynomial functions like f (x) � x3 � x2 � 3x � 3.
Since f (1) � �4 and f (2) � 3, there is at least one real zero between 1 and 2.
The midpoint of this interval is �1 �2
2� � 1.5. Since f(1.5) � �1.875, the zero is
between 1.5 and 2. The midpoint of this interval is �1.52� 2� � 1.75. Since
f(1.75) is about 0.172, the zero is between 1.5 and 1.75. The midpoint of this
interval is �1.5 �2
1.75� � 1.625 and f(1.625) is about �0.94. The zero is between
1.625 and 1.75. The midpoint of this interval is �1.6252� 1.75� � 1.6875. Since
f (1.6875) is about �0.41, the zero is between 1.6875 and 1.75. Therefore, the zero is 1.7 to the nearest tenth.
The diagram below summarizes the results obtained by the bisection method.
Using the bisection method, approximate to the nearest tenth the zero between the two integral values of x for each function.
1. f (x) � x3 � 4x2 � 11x � 2, f (0) � 2, f (1) � �12
2. f (x) � 2x4 � x2 � 15, f (1) � �12, f (2) � 21
3. f(x) � x5 � 2x3 � 12, f (1) � �13, f (2) � 4
4. f (x) � 4x3 � 2x � 7, f (�2) � �21, f (�1) � 5
5. f (x) � 3x3 � 14x2 � 27x � 126, f (4) � �14, f (5) � 16
1 1.5 21.625 1.75
1.6875
+ +––––sign of f (x ):
value x :
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
Study Guide and InterventionRational Zero Theorem
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
© Glencoe/McGraw-Hill 405 Glencoe Algebra 2
Less
on
7-6
Identify Rational Zeros
Rational Zero Let f(x) � a0xn � a1xn � 1 � … � an � 2x2 � an � 1x � an represent a polynomial function Theorem with integral coefficients. If �
pq� is a rational number in simplest form and is a zero of y � f(x),
then p is a factor of an and q is a factor of a0.
Corollary (Integral If the coefficients of a polynomial are integers such that a0 � 1 and an 0, any rational Zero Theorem) zeros of the function must be factors of an.
List all of the possible rational zeros of each function.
a. f(x) � 3x4 � 2x2 � 6x � 10
If �pq� is a rational root, then p is a factor of �10 and q is a factor of 3. The possible values
for p are 1, 2, 5, and 10. The possible values for q are 1 and 3. So all of the possible rational zeros are �
pq� � 1, 2, 5, 10, �
13�, �
23�, �
53�, and �
130�.
b. q(x) � x3 � 10x2 � 14x � 36
Since the coefficient of x3 is 1, the possible rational zeros must be the factors of theconstant term �36. So the possible rational zeros are 1, 2, 3, 4, 6, 9, 12, 18,and 36.
List all of the possible rational zeros of each function.
1. f(x) � x3 � 3x2 � x � 8 2. g(x) � x5 � 7x4 � 3x2 � x � 20
�1, �2, �4, �8 �1, �2, �4, �5, �10, �20
3. h(x) � x4 � 7x3 � 4x2 � x � 49 4. p(x) � 2x4 � 5x3 � 8x2 � 3x � 5
�1, �7, �49 �1, �5, � , �
5. q(x) � 3x4 � 5x3 � 10x � 12 6. r(x) � 4x5 � 2x � 18�1, �2, �3, �4, �6, �12, �1, �2, �3, �6, �9, �18,
� , � , � � , � , � , � , � , �
7. f(x) � x7 � 6x5 � 3x4 � x3 � 4x2 � 120 8. g(x) � 5x6 � 3x4 � 5x3 � 2x2 � 15
�1, �2, �3, �4, �5, �6, �8, �10, �12, �15, �20, �24, �30, �40, �60, �120
�1, �3, �5, �15, � , �
9. h(x) � 6x5 � 3x4 � 12x3 � 18x2 � 9x � 21 10. p(x) � 2x7 � 3x6 � 11x5 � 20x2 � 11
�1, �3, �7, �21, � , � , � , � , �1, �11, � , �
� , � , � , � 7�
1�
7�
1�
11�
1�
21�
7�
3�
1�
3�
1�
9�
3�
1�
9�
3�
1�
4�
2�
1�
5�
1�
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 406 Glencoe Algebra 2
Find Rational Zeros
Find all of the rational zeros of f(x) � 5x3 � 12x2 � 29x � 12.From the corollary to the Fundamental Theorem of Algebra, we know that there are exactly 3 complex roots. According to Descartes’ Rule of Signs there are 2 or 0 positive real roots and 1 negative real root. The possible rational zeros are 1, 2, 3, 4, 6, 12, , , , , , . Make a table and test some possible rational zeros.
Since f(1) � 0, you know that x � 1 is a zero.The depressed polynomial is 5x2 � 17x � 12, which can be factored as (5x � 3)(x � 4).By the Zero Product Property, this expression equals 0 when x � or x � �4.The rational zeros of this function are 1, , and �4.
Find all of the zeros of f(x) � 8x4 � 2x3 � 5x2 � 2x � 3.There are 4 complex roots, with 1 positive real root and 3 or 1 negative real roots. The possible rational zeros are 1, 3, , , , , , and .3
�83�4
3�2
1�8
1�4
1�2
3�5
3�5
�pq� 5 12 �29 12
1 5 17 �12 0
12�5
6�5
4�5
3�5
2�5
1�5
Study Guide and Intervention (continued)
Rational Zero Theorem
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
Example 1Example 1
Example 2Example 2
ExercisesExercises
Make a table and test some possible values.
Since f� � � 0, we know that x �
is a root.
1�2
1�2
�pq� 8 2 5 2 �3
1 8 10 15 17 14
2 8 18 41 84 165
�12
� 8 6 8 6 0
The depressed polynomial is 8x3 � 6x2 � 8x � 6.Try synthetic substitution again. Any remainingrational roots must be negative.
x � ��34� is another rational root.
The depressed polynomial is 8x2 � 8 � 0,which has roots i.
�pq� 8 6 8 6
��14
� 8 4 7 4�14
�
��34
� 8 0 8 0
The zeros of this function are �12�, ��
34�, and i.
Find all of the rational zeros of each function.
1. f(x) � x3 � 4x2 � 25x � 28 �1, 4, �7 2. f(x) � x3 � 6x2 � 4x � 24 �6
Find all of the zeros of each function.
3. f(x) � x4 � 2x3 � 11x2 � 8x � 60 4. f(x) � 4x4 � 5x3 � 30x2 � 45x � 54
3, �5, �2i , �2, �3i3�
Skills PracticeRational Zero Theorem
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
© Glencoe/McGraw-Hill 407 Glencoe Algebra 2
Less
on
7-6
List all of the possible rational zeros of each function.
1. n(x) � x2 � 5x � 3 2. h(x) � x2 � 2x � 5
�1, �3 �1, �5
3. w(x) � x2 � 5x � 12 4. f(x) � 2x2 � 5x � 3
�1, �2, �3, �4, �6, �12 ��12
�, ��32
�, �1, �3
5. q(x) � 6x3 � x2 � x � 2 6. g(x) � 9x4 � 3x3 � 3x2 � x � 27
��16
�, ��13
�, ��12
�, ��23
�, �1, �2 ��19
�, ��13
�, �1, �3, �9, �27
Find all of the rational zeros of each function.
7. f(x) � x3 � 2x2 � 5x � 4 � 0 8. g(x) � x3 � 3x2 � 4x � 12
1 �2, 2, 3
9. p(x) � x3 � x2 � x � 1 10. z(x) � x3 � 4x2 � 6x � 4
1 2
11. h(x) � x3 � x2 � 4x � 4 12. g(x) � 3x3 � 9x2 � 10x � 8
1 4
13. g(x) � 2x3 � 7x2 � 7x � 12 14. h(x) � 2x3 � 5x2 � 4x � 3
�4, �1, �32
� �1, �12
�, 3
15. p(x) � 3x3 � 5x2 � 14x � 4 � 0 16. q(x) � 3x3 � 2x2 � 27x � 18
��13
� ��23
�
17. q(x) � 3x3 � 7x2 � 4 18. f(x) � x4 � 2x3 � 13x2 � 14x � 24
��23
�, 1, 2 �3, �1, 2, 4
19. p(x) � x4 � 5x3 � 9x2 � 25x � 70 20. n(x) � 16x4 � 32x3 � 13x2 � 29x � 6
�2, 7 �1, �14
�, �34
�, 2
Find all of the zeros of each function.
21. f(x) � x3 � 5x2 � 11x � 15 22. q(x) � x3 � 10x2 � 18x � 4
�3, �1 � 2i, �1 � 2i 2, 4 � �14�, 4 � �14�
23. m(x) � 6x4 � 17x3 � 8x2 � 8x � 3 24. g(x) � x4 � 4x3 � 5x2 � 4x � 4
�13
�, �32
�, , �2, �2, �i, i1 � �5��
1 � �5��
© Glencoe/McGraw-Hill 408 Glencoe Algebra 2
List all of the possible rational zeros of each function.
1. h(x) � x3 � 5x2 � 2x � 12 2. s(x) � x4 � 8x3 � 7x � 14
�1, �2, �3, �4, �6, �12 �1, �2, �7, �14
3. f(x) � 3x5 � 5x2 � x � 6 4. p(x) � 3x2 � x � 7
��13
�, ��23
�, �1, �2, �3, �6 ��13
�, ��73
�, �1, �7
5. g(x) � 5x3 � x2 � x � 8 6. q(x) � 6x5 � x3 � 3
��15
�, ��25
�, ��45
�, ��85
�, �1, �2, �4, �8 ��16
�, ��13
�, ��12
�, ��32
�, �1, �3
Find all of the rational zeros of each function.
7. q(x) � x3 � 3x2 � 6x � 8 � 0 �4, �1, 2 8. v(x) � x3 � 9x2 � 27x � 27 3
9. c(x) � x3 � x2 � 8x � 12 �3, 2 10. f(x) � x4 � 49x2 0, �7, 7
11. h(x) � x3 � 7x2 � 17x � 15 3 12. b(x) � x3 � 6x � 20 �2
13. f(x) � x3 � 6x2 � 4x � 24 6 14. g(x) � 2x3 � 3x2 � 4x � 4 �2
15. h(x) � 2x3 � 7x2 � 21x � 54 � 0�3, 2, �
92
�16. z(x) � x4 � 3x3 � 5x2 � 27x � 36 �1, 4
17. d(x) � x4 � x3 � 16 no rational zeros 18. n(x) � x4 � 2x3 � 3 �1
19. p(x) � 2x4 � 7x3 � 4x2 � 7x � 6 20. q(x) � 6x4 � 29x3 � 40x2 � 7x � 12
�1, 1, �32
�, 2 ��32
�, ��43
�
Find all of the zeros of each function.
21. f(x) � 2x4 � 7x3 � 2x2 � 19x � 12 22. q(x) � x4 � 4x3 � x2 � 16x � 20
�1, �3, , �2, 2, 2 � i, 2 � i
23. h(x) � x6 � 8x3 24. g(x) � x6 � 1�1, 1, ,
0, 2, �1 � i�3�, �1 � i�3� , ,
25. TRAVEL The height of a box that Joan is shipping is 3 inches less than the width of thebox. The length is 2 inches more than twice the width. The volume of the box is 1540 in3.What are the dimensions of the box? 22 in. by 10 in. by 7 in.
26. GEOMETRY The height of a square pyramid is 3 meters shorter than the side of its base.If the volume of the pyramid is 432 m3, how tall is it? Use the formula V � �
13�Bh. 9 m
1 � i�3���
1 � i�3���
�1 � i�3���
�1 � i�3���
1 � �33���
1 � �33���
Practice (Average)
Rational Zero Theorem
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
Reading to Learn MathematicsRational Zero Theorem
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
© Glencoe/McGraw-Hill 409 Glencoe Algebra 2
Less
on
7-6
Pre-Activity How can the Rational Zero Theorem solve problems involving largenumbers?
Read the introduction to Lesson 7-6 at the top of page 378 in your textbook.
Rewrite the polynomial equation w(w � 8)(w � 5) � 2772 in the form f(x) � 0, where f(x) is a polynomial written in descending powers of x.w3 � 3w2 � 40w � 2772 � 0
Reading the Lesson
1. For each of the following polynomial functions, list all the possible values of p, all the possible values of q, and all the possible rational zeros �
pq�.
a. f(x) � x3 � 2x2 � 11x � 12
possible values of p: �1, �2, �3, �4, �6, �12
possible values of q: �1
possible values of �pq�: �1, �2, �3, �4, �6, �12
b. f(x) � 2x4 � 9x3 � 23x2 � 81x � 45
possible values of p: �1, �3, �5, �9, �15, �45
possible values of q: �1, �2
possible values of �pq�: �1, �3, �5, �9, �15, �45, ��
12
�, ��32
�, ��52
�, ��92
�, ��125�,
��425�
2. Explain in your own words how Descartes’ Rule of Signs, the Rational Zero Theorem, andsynthetic division can be used together to find all of the rational zeros of a polynomialfunction with integer coefficients.
Sample answer: Use Descartes’ Rule to find the possible numbers ofpositive and negative real zeros. Use the Rational Zero Theorem to listall possible rational zeros. Use synthetic division to test which of thenumbers on the list of possible rational zeros are actually zeros of thepolynomial function. (Descartes’ Rule may help you to limit thepossibilities.)
Helping You Remember
3. Some students have trouble remembering which numbers go in the numerators and whichgo in the denominators when forming a list of possible rational zeros of a polynomialfunction. How can you use the linear polynomial equation ax � b � 0, where a and b arenonzero integers, to remember this?Sample answer: The solution of the equation is ��
ba
�. The numerator b is a factor of the constant term in ax � b. The denominator a is a factor
© Glencoe/McGraw-Hill 410 Glencoe Algebra 2
Infinite Continued FractionsSome infinite expressions are actually equal to realnumbers! The infinite continued fraction at the right isone example.
If you use x to stand for the infinite fraction, then theentire denominator of the first fraction on the right isalso equal to x. This observation leads to the followingequation:
x � 1 � �1x�
Write a decimal for each continued fraction.
1. 1 � �11� 2. 1 � 3. 1 �
4. 1 � 5. 1 �
6. The more terms you add to the fractions above, the closer their value approaches the value of the infinite continued fraction. What value do the fractions seem to be approaching?
7. Rewrite x � 1 � �1x� as a quadratic equation and solve for x.
8. Find the value of the following infinite continued fraction.
3 � 1
3 � 1
3 � 1
3 � 13 � …
1
1 � 1
1 � 1
1 � 1
1 � 11
1
1 � 1
1 � 1
1 � 11
1
1 � 1
1 � 11
1
1 � 11
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
x � 1 �1
1 � 1
1 � 1
1 � 11 � …
Study Guide and InterventionOperations on Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
© Glencoe/McGraw-Hill 411 Glencoe Algebra 2
Less
on
7-7
Arithmetic Operations
Sum (f � g)(x) � f(x) � g(x)Difference (f � g)(x) � f(x) � g(x)
Operations with Functions Product (f � g)(x) � f(x) � g(x)
Quotient � �(x) � , g(x) 0
Find (f � g)(x), (f � g)(x), (f g)(x), and � �(x) for f(x) � x2 � 3x � 4and g(x) � 3x � 2.(f � g)(x) � f(x) � g(x) Addition of functions
� (x2 � 3x � 4) � (3x � 2) f(x) � x2 � 3x � 4, g(x) � 3x � 2
� x2 � 6x � 6 Simplify.
(f � g)(x) � f(x) � g(x) Subtraction of functions
� (x2 � 3x � 4) � (3x � 2) f(x) � x2 � 3x � 4, g(x) � 3x � 2
� x2 � 2 Simplify.
(f � g)(x) � f(x) � g(x) Multiplication of functions
� (x2 � 3x � 4)(3x � 2) f(x) � x2 � 3x � 4, g(x) � 3x � 2
� x2(3x � 2) � 3x(3x � 2) � 4(3x � 2) Distributive Property
� 3x3 � 2x2 � 9x2 � 6x � 12x � 8 Distributive Property
� 3x3 � 7x2 � 18x � 8 Simplify.
� �(x) � Division of functions
� , x �23� f(x) � x2 � 3x � 4 and g(x) � 3x � 2
Find (f � g)(x), (f � g)(x), (f g)(x), and � �(x) for each f(x) and g(x).
1. f(x) � 8x � 3; g(x) � 4x � 5 2. f(x) � x2 � x � 6; g(x) � x � 2
12x � 2; 4x � 8; 32x2 � 28x � 15; x2 � 2x � 8; x2 � 4;
, x � x3 � x2 � 8x � 12; x � 3, x 2
3. f(x) � 3x2 � x � 5; g(x) � 2x � 3 4. f(x) � 2x � 1; g(x) � 3x2 � 11x � 4
3x2 � x � 2; 3x2 � 3x � 8; 3x2 � 13x � 5; �3x2 � 9x � 3; 6x3 � 11x2 � 13x � 15; 6x3 � 19x2 � 19x � 4;
, x , x , �4
5. f(x) � x2 � 1; g(x) �
x2 � 1 � ; x2 � 1 � ; x � 1; x3 � x2 � x � 1, x �11�
1�
1�x � 1
1�
2x � 1��
3�
3x2 � x � 5��
5�
8x � 3�
f�g
x2 � 3x � 4��3x � 2
f(x)�g(x)
f�g
f�g
f(x)�g(x)
f�g
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 412 Glencoe Algebra 2
Composition of Functions
Composition Suppose f and g are functions such that the range of g is a subset of the domain of f.of Functions Then the composite function f � g can be described by the equation [f � g](x) � f [g (x)].
For f � {(1, 2), (3, 3), (2, 4), (4, 1)} and g � {(1, 3), (3, 4), (2, 2), (4, 1)},find f � g and g � f if they exist.f[ g(1)] � f(3) � 3 f[ g(2)] � f(2) � 4 f[ g(3)] � f(4) � 1 f[ g(4)] � f(1) � 2f � g � {(1, 3), (2, 4), (3, 1), (4, 2)}g[f(1)] � g(2) � 2 g[f(2)] � g(4) � 1 g[f(3)] � g(3) � 4 g[f(4)] � g(1) � 3g � f � {(1, 2), (2, 1), (3, 4), (4, 3)}
Find [g � h](x) and [h � g](x) for g(x) � 3x � 4 and h(x) � x2 � 1.[g � h](x) � g[h(x)] [h � g](x) � h[ g(x)]
� g(x2 � 1) � h(3x � 4)� 3(x2 � 1) � 4 � (3x � 4)2 � 1� 3x2 � 7 � 9x2 � 24x � 16 � 1
� 9x2 � 24x � 15
For each set of ordered pairs, find f � g and g � f if they exist.
1. f � {(�1, 2), (5, 6), (0, 9)}, 2. f � {(5, �2), (9, 8), (�4, 3), (0, 4)},g � {(6, 0), (2, �1), (9, 5)} g � {(3, 7), (�2, 6), (4, �2), (8, 10)}f � g � {(2, 2), (6, 9), (9, 6)}; f � g does not exist; g � f � {(�1, �1), (0, 5), (5, 0)} g � f � {(�4, 7), (0, �2), (5, 6), (9, 10)}
Find [f � g](x) and [g � f](x).
3. f(x) � 2x � 7; g(x) � �5x � 1 4. f(x) � x2 � 1; g(x) � �4x2
[f � g](x) � �10x � 5, [f � g](x) � 16x4 � 1, [g � f ](x) � �10x � 36 [g � f ](x) � �4x4 � 8x2 � 4
5. f(x) � x2 � 2x; g(x) � x � 9 6. f(x) � 5x � 4; g(x) � 3 � x[f � g](x) � x2 � 16x � 63, [f � g](x) � 19 � 5x, [g � f ](x) � x2 � 2x � 9 [g � f ](x) � �1 � 5x
Study Guide and Intervention (continued)
Operations on Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
Example 1Example 1
Example 2Example 2
ExercisesExercises
Skills PracticeOperations on Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
© Glencoe/McGraw-Hill 413 Glencoe Algebra 2
Less
on
7-7
Find (f � g)(x), (f � g)(x), (f g)(x), and � �(x) for each f(x) and g(x).
1. f(x) � x � 5 2x � 1; 9; 2. f(x) � 3x � 1 5x � 2; x � 4; 6x 2 � 7x �3;
g(x) � x � 4�xx
��
54
�, x 4g(x) � 2x � 3 �
32xx
��
13
�, x �32
�
3. f(x) � x2 x2 � x � 4; x2 � x � 4; 4. f(x) � 3x2 �3x3
x� 5�, x 0; �3x3
x� 5�, x
0;
g(x) � 4 � x 4x2 � x3; , x 4 g(x) � �5x� 15x, x 0; �35
x3�, x 0
For each set of ordered pairs, find f � g and g � f if they exist.
5. f � {(0, 0), (4, �2)} 6. f � {(0, �3), (1, 2), (2, 2)}g � {(0, 4), (�2, 0), (5, 0)} g � {(�3, 1), (2, 0)}{(0, �2), (�2, 0), (5, 0)}; {(�3, 2), (2, �3)}; {(0, 4), (4, 0)} {(0, 1), (1, 0), (2, 0)}
7. f � {(�4, 3), (�1, 1), (2, 2)} 8. f � {(6, 6), (�3, �3), (1, 3)}g � {(1, �4), (2, �1), (3, �1)} g � {(�3, 6), (3, 6), (6, �3)}{(1, 3), (2, 1), (3, 1)}; {(�3, 6), (3, 6), (6, �3)};{(�4, �1), (�1, �4), (2, �1)} {(6, �3), (�3, 6), (1, 6)}
Find [g � h](x) and [h � g](x).
9. g(x) � 2x 2x � 4; 2x � 2 10. g(x) � �3x �12x � 3; �12x � 1h(x) � x � 2 h(x) � 4x � 1
11. g(x) � x � 6 x; x 12. g(x) � x � 3 x2 � 3; x2 � 6x � 9h(x) � x � 6 h(x) � x2
13. g(x) � 5x 5x2 � 5x � 5; 14. g(x) � x � 2 2x2 � 1; 2x2 � 8x � 5h(x) � x2 � x � 1 25x2 � 5x � 1 h(x) � 2x2 � 3
If f(x) � 3x, g(x) � x � 4, and h(x) � x2 � 1, find each value.
15. f[ g(1)] 15 16. g[h(0)] 3 17. g[f(�1)] 1
18. h[f(5)] 224 19. g[h(�3)] 12 20. h[f(10)] 899
x2�
f�g
x2 � x � 20;
© Glencoe/McGraw-Hill 414 Glencoe Algebra 2
Find (f � g)(x), (f � g)(x), (f g)(x), and ��gf��(x) for each f(x) and g(x).
1. f(x) � 2x � 1 2. f(x) � 8x2 3. f(x) � x2 � 7x � 12
g(x) � x � 3 g(x) � g(x) � x2 � 9
3x � 2; x � 4; �8x4
x�2
1�, x 0; 2x2 � 7x � 3; 7x � 21;
2x2 � 5x � 3; �8x4
x2� 1�, x 0; x4 � 7x3 � 3x2 � 63x � 108;
�2xx
��
31
�, x 3 8, x 0; 8x4, x 0 �xx
��
43
�, x �3
For each set of ordered pairs, find f � g and g � f if they exist.
4. f � {(�9, �1), (�1, 0), (3, 4)} 5. f � {(�4, 3), (0, �2), (1, �2)}g � {(0, �9), (�1, 3), (4, �1)} g � {(�2, 0), (3, 1)}{(0, �1), (�1, 4), (4, 0)}; {(�2, �2), (3, �2)}; {(�9, 3), (�1, �9), (3, �1)} {(�4, 1), (0, 0), (1, 0)}
6. f � {(�4, �5), (0, 3), (1, 6)} 7. f � {(0, �3), (1, �3), (6, 8)}g � {(6, 1), (�5, 0), (3, �4)} g � {(8, 2), (�3, 0), (�3, 1)}{(6, 6), (�5, 3), (3, �5)}; does not exist; {(�4, 0), (0, �4), (1, 1)} {(0, 0), (1, 0), (6, 2)}
Find [g � h](x) and [h � g](x).
8. g(x) � 3x 9. g(x) � �8x 10. g(x) � x � 6h(x) � x � 4 h(x) � 2x � 3 h(x) � 3x2 3x2 � 6;3x � 12; 3x � 4 �16x � 24; �16x � 3 3x2 � 36x � 108
11. g(x) � x � 3 12. g(x) � �2x 13. g(x) � x � 2h(x) � 2x2 h(x) � x2 � 3x � 2 h(x) � 3x2 � 12x2 � 3; �2x2 � 6x � 4; 3x2 � 1; 2x2 � 12x � 18 4x2 � 6x � 2 3x2 � 12x � 13
If f(x) � x2, g(x) � 5x, and h(x) � x � 4, find each value.
14. f[ g(1)] 25 15. g[h(�2)] 10 16. h[f(4)] 20
17. f[h(�9)] 25 18. h[ g(�3)] �11 19. g[f(8)] 320
20. h[f(20)] 404 21. [f � (h � g)](�1) 1 22. [f � (g � h)](4) 1600
23. BUSINESS The function f(x) � 1000 � 0.01x2 models the manufacturing cost per itemwhen x items are produced, and g(x) � 150 � 0.001x2 models the service cost per item.Write a function C(x) for the total manufacturing and service cost per item.C(x) � 1150 � 0.011x2
24. MEASUREMENT The formula f � �1n2� converts inches n to feet f, and m � �52
f80� converts
feet to miles m. Write a composition of functions that converts inches to miles.
[m � f ]n � �63,
n360�
1�x2
Practice (Average)
Operations on Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
Reading to Learn MathematicsOperations on Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
© Glencoe/McGraw-Hill 415 Glencoe Algebra 2
Less
on
7-7
Pre-Activity Why is it important to combine functions in business?
Read the introduction to Lesson 7-7 at the top of page 383 in your textbook.
Describe two ways to calculate Ms. Coffmon’s profit from the sale of 50 birdhouses. (Do not actually calculate her profit.) Sample answer: 1. Find the revenue by substituting 50 for x in the expression125x. Next, find the cost by substituting 50 for x in theexpression 65x � 5400. Finally, subtract the cost from therevenue to find the profit. 2. Form the profit function p(x) � r(x) � c(x) � 125x � (65x � 5400) � 60x � 5400.Substitute 50 for x in the expression 60x � 5400.
Reading the Lesson
1. Determine whether each statement is true or false. (Remember that true means always true.)
a. If f and g are polynomial functions, then f � g is a polynomial function. true
b. If f and g are polynomial functions, then is a polynomial function. false
c. If f and g are polynomial functions, the domain of the function f � g is the set of allreal numbers. true
d. If f(x) � 3x � 2 and g(x) � x � 4, the domain of the function is the set of all realnumbers. false
e. If f and g are polynomial functions, then (f � g)(x) � (g � f)(x). false
f. If f and g are polynomial functions, then (f � g)(x) � (g � f)(x) true
2. Let f(x) � 2x � 5 and g(x) � x2 � 1.
a. Explain in words how you would find ( f � g)(�3). (Do not actually do any calculations.)Sample answer: Square �3 and add 1. Take the number you get,multiply it by 2, and subtract 5.
b. Explain in words how you would find (g � f)(�3). (Do not actually do anycalculations.) Sample answer: Multiply �3 by 2 and subtract 5. Take thenumber you get, square it, and add 1.
Helping You Remember
3. Some students have trouble remembering the correct order in which to apply the twooriginal functions when evaluating a composite function. Write three sentences, each ofwhich explains how to do this in a slightly different way. (Hint: Use the word closest inthe first sentence, the words inside and outside in the second, and the words left andright in the third.) Sample answer: 1. The function that is written closest tothe variable is applied first. 2. Work from the inside to the outside. 3. Work from right to left.
f�g
f�g
© Glencoe/McGraw-Hill 416 Glencoe Algebra 2
Relative Maximum ValuesThe graph of f (x) � x3 � 6x � 9 shows a relative maximum value somewhere between f (�2) and f (�1). You can obtain a closer approximation by comparing values such as those shown in the table.
To the nearest tenth a relative maximum value for f (x) is �3.3.
Using a calculator to find points, graph each function. To the nearest tenth, find a relative maximum value of the function.
1. f (x) � x(x2 � 3) 2. f (x) � x3 � 3x � 3
3. f (x) � x3 � 9x � 2 4. f (x) � x3 � 2x2 � 12x � 24
5
x
f(x)
O 1
2
x
f(x)
O 2
x
f(x)
O
x
f(x)
O
x f (x)
�2 �5
�1.5 �3.375
�1.4 �3.344
�1.3 �3.397
�1 �4
x
f(x)
O 2–2–4
–8
–12
–16
–20
–4 4
f(x) � x3 � 6x � 9
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
Study Guide and InterventionInverse Functions and Relations
NAME ______________________________________________ DATE ____________ PERIOD _____
7-87-8
© Glencoe/McGraw-Hill 417 Glencoe Algebra 2
Less
on
7-8
Find Inverses
Inverse RelationsTwo relations are inverse relations if and only if whenever one relation contains the element (a, b), the other relation contains the element (b, a).
Property of Inverse Suppose f and f�1 are inverse functions.Functions Then f(a) � b if and only if f�1(b) � a.
Find the inverse of the function f(x) � x � . Then graph thefunction and its inverse.Step 1 Replace f(x) with y in the original equation.
f(x) � �25�x � → y � �
25�x �
Step 2 Interchange x and y.
x � �25�y �
Step 3 Solve for y.
x � �25�y � Inverse
5x � 2y � 1 Multiply each side by 5.
5x � 1 � 2y Add 1 to each side.
(5x � 1) � y Divide each side by 2.
The inverse of f(x) � �25�x � is f�1(x) � (5x � 1).
Find the inverse of each function. Then graph the function and its inverse.
1. f(x) � x � 1 2. f(x) � 2x � 3 3. f(x) � x � 2
f�1(x) � x � f�1(x) � x � f�1(x) � 4x � 8
x
f(x)
Ox
f(x)
O
x
f(x)
O
3�
1�
3�
3�
1�4
2�3
1�2
1�5
1�2
1�5
1�5
1�5
1�5
x
f(x)
O
f(x) � 2–5x � 1–5
f –1(x) � 5–2x � 1–2
1�5
2�5
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 418 Glencoe Algebra 2
Inverses of Relations and Functions
Inverse Functions Two functions f and g are inverse functions if and only if [f � g](x) � x and [g � f ](x) � x.
Determine whether f(x) � 2x � 7 and g(x) � (x � 7) are inversefunctions.
[ f � g](x) � f[ g(x)] [ g � f ](x) � g[ f(x)]
� f��12�(x � 7)� � g(2x � 7)
� 2��12�(x � 7)� � 7 � �
12�(2x � 7 � 7)
� x � 7 � 7 � x� x
The functions are inverses since both [ f � g](x) � x and [ g � f ](x) � x.
Determine whether f(x) � 4x � and g(x) � x � 3 are inversefunctions.
[ f � g](x) � f[ g(x)]
� f��14�x � 3�
� 4��14�x � 3� � �
13�
� x � 12 � �13�
� x � 11�23�
Since [ f � g](x) x, the functions are not inverses.
Determine whether each pair of functions are inverse functions.
1. f(x) � 3x � 1 2. f(x) � �14�x � 5 3. f(x) � �
12�x � 10
g(x) � �13�x � �
13� yes g(x) � 4x � 20 yes g(x) � 2x � �1
10� no
4. f(x) � 2x � 5 5. f(x) � 8x � 12 6. f(x) � �2x � 3
g(x) � 5x � 2 no g(x) � �18�x � 12 no g(x) � ��
12�x � �
32� yes
7. f(x) � 4x � �12� 8. f(x) � 2x � �
35� 9. f(x) � 4x � �
12�
g(x) � �14�x � �
18� yes g(x) � �1
10�(5x � 3) yes g(x) � �
12�x � �
32� no
10. f(x) � 10 � �2x
� 11. f(x) � 4x � �45� 12. f(x) � 9 � �
32�x
g(x) � 20 � 2x yes g(x) � �4x
� � �15� yes g(x) � �
23�x � 6 yes
1�4
1�3
1�2
Study Guide and Intervention (continued)
Inverse Functions and Relations
NAME ______________________________________________ DATE ____________ PERIOD _____
7-87-8
Example 1Example 1
Example 2Example 2
ExercisesExercises
Skills PracticeInverse Functions and Relations
NAME ______________________________________________ DATE ____________ PERIOD _____
7-87-8
© Glencoe/McGraw-Hill 419 Glencoe Algebra 2
Less
on
7-8
Find the inverse of each relation.
1. {(3, 1), (4, �3), (8, �3)} 2. {(�7, 1), (0, 5), (5, �1)}{(1, 3), (�3, 4), (�3, 8)} {(1, �7), (5, 0), (�1, 5)}
3. {(�10, �2), (�7, 6), (�4, �2), (�4, 0)} 4. {(0, �9), (5, �3), (6, 6), (8, �3)}{(�2, �10), (6, �7), (�2, �4), (0, �4)} {(�9, 0), (�3, 5), (6, 6), (�3, 8)}
5. {(�4, 12), (0, 7), (9, �1), (10, �5)} 6. {(�4, 1), (�4, 3), (0, �8), (8, �9)}{(12, �4), (7, 0), (�1, 9), (�5, 10)} {(1, �4), (3, �4), (�8, 0), (�9, 8)}
Find the inverse of each function. Then graph the function and its inverse.
7. y � 4 8. f(x) � 3x 9. f(x) � x � 2
x � 4 f �1(x) � �13
�x f �1(x) � x � 2
10. g(x) � 2x � 1 11. h(x) � �14�x 12. y � �
23�x � 2
g�1(x) � �x �
21
� h�1(x) � 4x y � �32
�x � 3
Determine whether each pair of functions are inverse functions.
13. f(x) � x � 1 no 14. f(x) � 2x � 3 yes 15. f(x) � 5x � 5 yesg(x) � 1 � x g(x) � �
12�(x � 3) g(x) � �
15�x � 1
16. f(x) � 2x yes 17. h(x) � 6x � 2 no 18. f(x) � 8x � 10 yesg(x) � �
12�x g(x) � �
16�x � 3 g(x) � �
18�x � �
54�
x
y
Ox
h(x)
Ox
g(x)
O
x
f(x)
Ox
f(x)
Ox
y
O
© Glencoe/McGraw-Hill 420 Glencoe Algebra 2
Find the inverse of each relation.
1. {(0, 3), (4, 2), (5, �6)} 2. {(�5, 1), (�5, �1), (�5, 8)}{(3, 0), (2, 4), (�6, 5)} {(1, �5), (�1, �5), (8, �5)}
3. {(�3, �7), (0, �1), (5, 9), (7, 13)} 4. {(8, �2), (10, 5), (12, 6), (14, 7)}{(�7, �3), (�1, 0), (9, 5), (13, 7)} {(�2, 8), (5, 10), (6, 12), (7, 14)}
5. {(�5, �4), (1, 2), (3, 4), (7, 8)} 6. {(�3, 9), (�2, 4), (0, 0), (1, 1)}{(�4, �5), (2, 1), (4, 3), (8, 7)} {(9, �3), (4, �2), (0, 0), (1, 1)}
Find the inverse of each function. Then graph the function and its inverse.
7. f(x) � �34�x 8. g(x) � 3 � x 9. y � 3x � 2
f�1(x) � �43
�x g�1(x) � x � 3 y � �x �
32
�
Determine whether each pair of functions are inverse functions.
10. f(x) � x � 6 yes 11. f(x) � �4x � 1 yes 12. g(x) � 13x � 13 nog(x) � x � 6 g(x) � �
14�(1 � x) h(x) � �1
13�x � 1
13. f(x) � 2x no 14. f(x) � �67�x yes 15. g(x) � 2x � 8 yes
g(x) � �2x g(x) � �76�x h(x) � �
12�x � 4
16. MEASUREMENT The points (63, 121), (71, 180), (67, 140), (65, 108), and (72, 165) givethe weight in pounds as a function of height in inches for 5 students in a class. Give thepoints for these students that represent height as a function of weight.(121, 63), (180, 71), (140, 67), (108, 65), (165, 72)
REMODELING For Exercises 17 and 18, use the following information.The Clearys are replacing the flooring in their 15 foot by 18 foot kitchen. The new flooringcosts $17.99 per square yard. The formula f(x) � 9x converts square yards to square feet.
17. Find the inverse f�1(x). What is the significance of f�1(x) for the Clearys? f�1(x) � �x9
�; It will allow them to convert the square footage of their kitchen floor tosquare yards, so they can then calculate the cost of the new flooring.
18. What will the new flooring cost the Cleary’s? $539.70
x
g(x)
Ox
f(x)
O
Practice (Average)
Inverse Functions and Relations
NAME ______________________________________________ DATE ____________ PERIOD _____
7-87-8
Reading to Learn MathematicsInverse Functions and Relations
NAME ______________________________________________ DATE ____________ PERIOD _____
7-87-8
© Glencoe/McGraw-Hill 421 Glencoe Algebra 2
Less
on
7-8
Pre-Activity How are inverse functions related to measurement conversions?
Read the introduction to Lesson 7-8 at the top of page 390 in your textbook.
A function multiplies a number by 3 and then adds 5 to the result. What doesthe inverse function do, and in what order? Sample answer: It firstsubtracts 5 from the number and then divides the result by 3.
Reading the Lesson
1. Complete each statement.
a. If two relations are inverses, the domain of one relation is the ofthe other.
b. Suppose that g(x) is a relation and that the point (4, �2) is on its graph. Then a point
on the graph of g�1(x) is .
c. The test can be used on the graph of a function to determine
whether the function has an inverse function.
d. If you are given the graph of a function, you can find the graph of its inverse by
reflecting the original graph over the line with equation .
e. If f and g are inverse functions, then (f � g)(x) � and
(g � f)(x) � .
f. A function has an inverse that is also a function only if the given function is
.
g. Suppose that h(x) is a function whose inverse is also a function. If h(5) � 12, thenh�1(12) � .
2. Assume that f(x) is a one-to-one function defined by an algebraic equation. Write the foursteps you would follow in order to find the equation for f�1(x).
1. Replace f(x) with y in the original equation.
2. Interchange x and y.
3. Solve for y.
4. Replace y with f �1(x).
Helping You Remember
3. A good way to remember something new is to relate it to something you already know.How are the vertical and horizontal line tests related? Sample answer: The verticalline test determines whether a relation is a function because the orderedpairs in a function can have no repeated x-values. The horizontal linetest determines whether a function is one-to-one because a one-to-onefunction cannot have any repeated y-values.
5
one-to-one
xx
y � x
horizontal line
(�2, 4)
range
© Glencoe/McGraw-Hill 422 Glencoe Algebra 2
Miniature GolfIn miniature golf, the object of the game is to roll the golf ball into the hole in as few shots as possible. As in the diagram at the right,the hole is often placed so that a direct shot is impossible. Reflectionscan be used to help determine the direction that the ball should berolled in order to score a hole-in-one.
Using wall E�F�, find the path to use to score a hole-in-one.
Find the reflection image of the “hole” with respect to E�F� and label it H . The intersection of B�H� � with wall E�F� is the point at which the shot should be directed.
For the hole at the right, find a path to score a hole-in-one.
Find the reflection image of H with respect to E�F� and label it H .In this case, B�H� � intersects J�K� before intersecting E�F�. Thus, thispath cannot be used. To find a usable path, find the reflection image of H with respect to G�F� and label it H�. Now, the intersection of B�H��� with wall G�F� is the point at which the shotshould be directed.
Copy each figure. Then, use reflections to determine a possible path for a hole-in-one.
1. 2. 3.
B
G F
J K H'
H"
E
H
Ball
Hole
E
H'
F
Ball
Hole
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-87-8
Example 1Example 1
Example 2Example 2
Study Guide and InterventionSquare Root Functions and Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
7-97-9
© Glencoe/McGraw-Hill 423 Glencoe Algebra 2
Less
on
7-9
Square Root Functions A function that contains the square root of a variableexpression is a square root function.
Graph y � �3x ��2�. State its domain and range.
Since the radicand cannot be negative, 3x � 2 � 0 or x � �23�.
The x-intercept is �23�. The range is y � 0.
Make a table of values and graph the function.
Graph each function. State the domain and range of the function.
1. y � �2x� 2. y � �3�x� 3. y � ����2x�
D: x � 0; R: y � 0 D: x � 0; R: y � 0 D: x � 0; R: y � 0
4. y � 2�x � 3� 5. y � ��2x � 3� 6. y � �2x � 5�
D: x � 3; R: y � 0 D: x � �32
�; R: y � 0 D: x � ��52
�; R: y � 0
x
y
O
x
y
O
x
y
O
x
y
O
xy
O
x
y
O
x y
�23
� 0
1 1
2 2
3 �7�
x
y
O
y � ����3x � 2
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 424 Glencoe Algebra 2
Square Root Inequalities A square root inequality is an inequality that containsthe square root of a variable expression. Use what you know about graphing square rootfunctions and quadratic inequalities to graph square root inequalities.
Graph y � �2x ��1� � 2.Graph the related equation y � �2x � 1� � 2. Since the boundary should be included, the graph should be solid.
The domain includes values for x � �12�, so the graph is to the right
of x � �12�. The range includes only numbers greater than 2, so the
graph is above y � 2.
Graph each inequality.
1. y � 2�x� 2. y � �x � 3� 3. y � 3�2x � 1�
4. y � �3x � 4� 5. y � �x � 1� � 4 6. y � 2�2x � 3�
7. y � �3x � 1� � 2 8. y � �4x � 2� � 1 9. y � 2�2x � 1� � 4
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
x
y
Ox
y
O
x
y
O
y � ����2x � 1 � 2
Study Guide and Intervention (continued)
Square Root Functions and Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
7-97-9
ExampleExample
ExercisesExercises
Skills PracticeSquare Root Functions and Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
7-97-9
© Glencoe/McGraw-Hill 425 Glencoe Algebra 2
Less
on
7-9
Graph each function. State the domain and range of each function.
1. y � �2x� 2. y � ��3x� 3. y � 2�x�
D: x � 0, R: y � 0 D: x � 0, R: y � 0 D: x � 0, R: y � 0
4. y � �x � 3� 5. y � ��2x � 5� 6. y � �x � 4� � 2
D: x � �3, R: y � 0 D: x � 2.5, R: y � 0 D: x � �4, R: y � �2
Graph each inequality.
7. y � �4x� 8. y � �x � 1� 9. y � �4x � 3�
x
y
Ox
y
Ox
y
O
x
y
Ox
y
O
x
y
O
x
y
O
x
y
Ox
y
O
© Glencoe/McGraw-Hill 426 Glencoe Algebra 2
Graph each function. State the domain and range of each function.
1. y � �5x� 2. y � ��x � 1� 3. y � 2�x � 2�
D: x � 0, R: y � 0 D: x � 1, R: y � 0 D: x � �2, R: y � 0
4. y � �3x � 4� 5. y � �x � 7� � 4 6. y � 1 � �2x � 3�
D: x � �43
�, R: y � 0 D: x � �7, R: y � �4 D: x � ��32
�, R: y � 1
Graph each inequality.
7. y � ��6x� 8. y � �x � 5� � 3 9. y � �2�3x � 2�
10. ROLLER COASTERS The velocity of a roller coaster as it moves down a hill is v � �v0
2 ��64h�, where v0 is the initial velocity and h is the vertical drop in feet. If v � 70 feet per second and v0 � 8 feet per second, find h. about 75.6 ft
11. WEIGHT Use the formula d � �� � 3960, which relates distance from Earth d
in miles to weight. If an astronaut’s weight on Earth WE is 148 pounds and in space Ws is115 pounds, how far from Earth is the astronaut? about 532 mi
39602 WE��Ws
x
y
O
x
y
O
x
y
O
x
y
O
x
y
O
Practice (Average)
Square Root Functions and Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
7-97-9
Reading to Learn MathematicsSquare Root Functions
NAME ______________________________________________ DATE ____________ PERIOD _____
7-97-9
© Glencoe/McGraw-Hill 427 Glencoe Algebra 2
Less
on
7-9
Pre-Activity How are square root functions used in bridge design?
Read the introduction to Lesson 7-9 at the top of page 395 in your textbook.
If the weight to be supported by a steel cable is doubled, should thediameter of the support cable also be doubled? If not, by what numbershould the diameter be multiplied?
no; �2�
Reading the Lesson
1. Match each square root function from the list on the left with its domain and range fromthe list on the right.
a. y � �x� iv i. domain: x � 0; range: y � 3
b. y � �x � 3� viii ii. domain: x � 0; range: y � 0
c. y � �x� � 3 i iii. domain: x � 0; range: y � �3
d. y � �x � 3� v iv. domain: x � 0; range: y � 0
e. y � ��x� ii v. domain: x � 3; range: y � 0
f. y � ��x � 3� vii vi. domain: x � 3; range: y � 3
g. y � �3 � x� � 3 vi vii. domain: x � 3; range: y � 0
h. y � ��x� � 3 iii viii. domain: x � �3; range: y � 0
2. The graph of the inequality y � �3x � 6� is a shaded region. Which of the followingpoints lie inside this region?
(3, 0) (2, 4) (5, 2) (4, �2) (6, 6)
(3, 0), (5, 2), (4, �2)
Helping You Remember
3. A good way to remember something is to explain it to someone else. Suppose you arestudying this lesson with a classmate who thinks that you cannot have square rootfunctions because every positive real number has two square roots. How would youexplain the idea of square root functions to your classmate?
Sample answer: To form a square root function, choose either the positive or negative square root. For example, y � �x� and y � ��x� aretwo separate functions.
© Glencoe/McGraw-Hill 428 Glencoe Algebra 2
Reading AlgebraIf two mathematical problems have basic structural similarities,they are said to be analogous. Using analogies is one way ofdiscovering and proving new theorems.
The following numbered sentences discuss a three-dimensionalanalogy to the Pythagorean theorem.
01 Consider a tetrahedron with three perpendicular faces thatmeet at vertex O.
02 Suppose you want to know how the areas A, B, and C of the three faces that meet at vertex O are related to the area Dof the face opposite vertex O.
03 It is natural to expect a formula analogous to the Pythagorean theorem z2 � x2 � y2, which is true for a similar situation in two dimensions.
04 To explore the three-dimensional case, you might guess a formula and then try to prove it.
05 Two reasonable guesses are D3 � A3 � B3 � C3 and D2 � A2 � B2 � C2.
Refer to the numbered sentences to answer the questions.
1. Use sentence 01 and the top diagram. The prefix tetra- means four. Write aninformal definition of tetrahedron.
2. Use sentence 02 and the top diagram. What are the lengths of the sides ofeach face of the tetrahedron?
3. Rewrite sentence 01 to state a two-dimensional analogue.
4. Refer to the top diagram and write expressions for the areas A, B, and C
5. To explore the three-dimensional case, you might begin by expressing a, b,and c in terms of p, q, and r. Use the Pythagorean theorem to do this.
6. Which guess in sentence 05 seems more likely? Justify your answer.
y
O
z
x
b
c
O
p
a
qr
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-97-9
Chapter 7 Test, Form 1
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 429 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. Find p(�3) if p(x) � 4 � x.A. 12 B. 4 C. 1 D. 7 1.
2. State the number of real zeros for the function whose graph is shown at the right.A. 0 B. 1C. 2 D. 3 2.
For Questions 3 and 4, use the graph shown at the right.
3. Determine the values of x between which a real zero is located.A. between �1 and 0B. between 6 and 7C. between �2 and �1D. between 2 and 3 3.
4. Estimate the x-coordinate at which a relative minimum occurs.A. 3 B. 2 C. 0 D. �1 4.
5. Write the expression x4 � 5x2 � 8 in quadratic form, if possible.A. (x2)2 � 5(x2) � 8 B. (x4)2 � 5(x4) � 8C. (x2)2 � 5(x2) � 8 D. not possible 5.
6. Solve x4 � 13x2 � 36 � 0.A. �3, �2, 2, 3 B. �9, �4, 4, 9 C. 2, 3, 2i, 3i D. �2, �3, 2i, 3i 6.
7. Use synthetic substitution to find f(3) for f(x) � x2 � 9x � 5.A. �23 B. �16 C. �13 D. 41 7.
8. One factor of x3 � 4x2 � 11x � 30 is x � 2. Find the remaining factors.A. x � 5, x � 3 B. x � 3, x � 5 C. x � 6, x � 5 D. x � 5, x � 6 8.
9. Which describes the number and type of roots of the equation 4x � 7 � 0?A. 1 imaginary root B. 1 real root and 1 imaginary rootC. 2 real roots D. 1 real root 9.
10. Which is not a root of the equation x3 � x2 � 10x � 8 � 0?A. 1 B. 4 C. �2 D. �1 10.
11. List all of the possible rational zeros of f(x) � x3 � 7x2 � 8x � 6.
A. �1, ��12�, � �
13�, �
16� B. 0, �1, �2, �3, �6
C. �1, �2, �3, �4, �6 D. �1, �2, �3, �6 11.
77
xO
f(x )
xO
f(x )
© Glencoe/McGraw-Hill 430 Glencoe Algebra 2
Chapter 7 Test, Form 1 (continued)
12. Find all of the rational zeros of p(x) � x3 � 12x � 16.A. �2, 4 B. 2, �4 C. 4 D. �2 12.
For Questions 13 and 14, use f(x) � x � 5 and g(x) � 2x.
13. Find (f � g)(x).A. 3x � 5 B. x � 5 C. 2x � 10 D. 2x2 � 5 13.
14. Find (f � g)(x).A. 2x2 � 5 B. 3x2 � 10x C. 2x2 � 10x D. 2x � 10 14.
15. If f(x) � 3x � 7 and g(x) � 2x � 5, find g[f(�3)].A. �26 B. �9 C. �1 D. 10 15.
16. If f(x) � x2 and g(x) � 3x � 1 find [ g � f](x).A. x2 � 3x � 1 B. 9x2 � 1C. 9x2 � 6x � 1 D. 3x2 � 1 16.
17. Find the inverse of g(x) � �3x.A. g�1(x) � x � 1 B. g�1(x) � �3x � 3
C. g�1(x) � x � 1 D. g�1(x) � ��13�x 17.
18. Determine which pair of functions are inverse functions.A. f(x) � x � 4 B. f(x) � x � 4
g(x) � x � 4 g(x) � 4x � 1C. f(x) � x � 4 D. f(x) � 4x � 1 18.
g(x) � �x �
44
� g(x) � 4x � 1
19. State the domain and range of the function graphed.A. D: x � 2, R: y � 0B. D: x 2, R: y � 0C. D: x 2, R: y 0D. D: x 2, R: y 0 19.
20. Which inequality is graphed?
A. y � �4x � 8�B. y � �4x � 8�C. y �4x � 8�D. y �4x � 8� 20.
Bonus If g(x) � 2x � 1, find g[g(x)]. B:
NAME DATE PERIOD
77
y
xO
y
xO
Chapter 7 Test, Form 2A
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 431 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. Find p(�4) if p(x) � 3x3 � 2x2 � 6x � 4.A. �188 B. �252 C. �140 D. �204 1.
2. If r(x) � x3 � 2x � 1, find r(2a3).A. 8a6 � 4a3 � 1 B. 4a6 � 4a3 � 1C. 6a6 � 4a3 � 1 D. 8a9 � 4a3 � 1 2.
3. State the number of real zeros for the function whose graph is shown at the right.A. 0 B. 2C. 3 D. 1 3.
For Questions 4 and 5, use the graph shown.
4. Determine the values of x between which a real zero is located.A. between 1 and 2B. between �4 and �3C. between �2 and �1D. between 2 and 3 4.
5. Estimate the x-coordinate at which a relative maximum occurs.A. 1 B. �1 C. 2 D. �2 5.
6. Write the expression 10x8 � 6x4 � 20 in quadratic form, if possible.A. 10(x4)2 � 6(x2)2 � 20 B. 10(x4)2 � 6(x4) � 20C. 10(x2)4 � 6(x2)2 � 20 D. not possible 6.
7. Solve x4 � 6x2 � 27 � 0.A. �3�, 3, 3i, i�3� B. �3, ��3�, �3�, 3C. �3, 3, i�3�, �i�3� D. ��3�, 3, 3i, �3i 7.
8. Use synthetic substitution to find f(�2) for f(x) � 2x4 � 3x3 � x2 � x � 5.A. 15 B. 67 C. 63 D. 19 8.
9. One factor of x3 � 3x2 � 4x � 12 is x � 2. Find the remaining factors.A. x � 2, x � 3 B. x � 2, x � 3 C. x � 2, x � 3 D. x � 2, x � 3 9.
10. Which describes the number and type of roots of the equation x4 � 64 � 0?A. 2 real roots, 2 imaginary roots B. 4 real rootsC. 3 real roots, 1 imaginary root D. 4 imaginary roots 10.
11. State the possible number of imaginary zeros of f(x) � 7x3 � x2 � 10x � 4.A. exactly 1 B. exactly 3 C. 3 or 1 D. 2 or 0 11.
77
xO
f(x )
xO
f(x )
© Glencoe/McGraw-Hill 432 Glencoe Algebra 2
Chapter 7 Test, Form 2A (continued)
12. Write a polynomial function of least degree with integral coefficients whose zeros include 4 and 2i.A. f(x) � x2 � 4 B. f(x) � x3 � 4x2 � 4x � 16C. f(x) � x3 � 4x2 � 4x � 16 D. f(x) � x3 � 4x2 � 4x � 16 12.
13. List all of the possible rational zeros of f(x) � 3x3 � 2x2 � 7x � 6.
A. �1, �2, �3, �6 B. 0, �1, �2, �3, �6, ��13�, ��
23�
C. �1, �2, �3, �6, ��13�, ��
23� D. �1, �3, ��
16�, ��
13�, ��
12�, ��
32� 13.
14. Find all of the rational zeros of f(x) � 4x3 � 3x2 � 22x � 15.
A. ��52�, �1, �3 B. �1, 3 C. �1, 3 D. �5, �1, 3 14.
15. Find ( f � g)(x) for f(x) � 3x2 and g(x) � 5 � x.A. 3x2 � x � 5 B. 75 � 30x � 3x2
C. 3x2 � 15x2 D. 15x2 � 3x3 15.
16. If f(x) � x2 � 1, and g(x) � x � 2, find [f � g](x).A. x2 � 4x � 5 B. x2 � 1C. x2 � 3 D. x3 � 2x2 � x � 2 16.
17. State the domain and range of the function graphed at the right.A. D: x � �3, R: y � 0B. D: x � �3, R: y 0C. D: x �3, R: y 0D. D: x �3, R: y � 0 17.
18. Find the inverse of f(x) � 2x � 7.
A. f�1(x) � 7x � 2 B. f�1(x) � �12�x � 7
C. f�1(x) � �x �
27
� D. f�1(x) � x � �72� 18.
19. Determine which pair of functions are inverse functions.A. f(x) � 3x � 1 B. f(x) � 2x � 5 C. f(x) � 2x � 2 D. f(x) � 3x � 8 19.
g(x) � �3x1� 1� g(x) � �
x �2
5� g(x) � 2x � 2 g(x) � �
13�x � 8
20. Which inequality is graphed at the right?A. y � �x � 4� B. y �x � 4�C. y �x � 4� D. y � �x � 4� 20.
Bonus If f(x) � 3x � 4, solve f [f(x)] � f(x) for x. B:
NAME DATE PERIOD
77
y
xO
2
4
2�2
y
xO
8
�8 4
�4
�4
Chapter 7 Test, Form 2B
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 433 Glencoe Algebra 2
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. Find p(�3) if p(x) � 4x3 � 5x2 � 7x � 10.A. �94 B. 32 C. �184 D. �142 1.
2. If r(x) � 4x2 � 3x � 7, find r(3a2).A. 36a4 � 9a2 � 7 B. 144a4 � 9a2 � 7C. 36a4 � 9a2 � 7 D. 12a4 � 9a2 � 7 2.
3. State the number of real zeros for the function whose graph is shown.A. 1 B. 4C. 3 D. 2 3.
For Questions 4 and 5, use the graph shown.
4. Determine the values of x between which a real zero is located.A. between �2 and �1B. between �1 and 0C. between 0 and 1D. between �3 and �2 4.
5. Estimate the x-coordinate at which a relative minimum occurs.A. �1 B. 0 C. 1 D. 2 5.
6. Write the expression 9n6 � 7n3 � 6 in quadratic form, if possible.A. 9(n3)3 � 7(n3) � 6 B. 9(n2)3 � 7(n2) � 6C. 9(n3)2 � 7(n3) � 6 D. not possible 6.
7. Solve b4 � 2b2 � 24 � 0.A. �2, ��6�, �6�, 2 B. ��6�, 2, 2i, i�6�C. �2, 2, �i�6�, i�6� D. �2i, 2i, ��6�, �6� 7.
8. Use synthetic substitution to find f(�3) for f(x) � x4 � 4x3 � 2x2 � 4x � 6.A. 9 B. 225 C. 201 D. �15 8.
9. One factor of x3 � 2x2 � 11x � 12 is x � 4. Find the remaining factors.A. x � 1, x � 3 B. x � 1, x � 3 C. x � 1, x � 3 D. x � 1, x � 3 9.
10. Which describes the number and type of roots of the equation x3 � 121x � 0?A. 1 real root, 2 imaginary roots B. 2 real roots, 1 imaginary rootC. 3 real roots D. 3 imaginary roots 10.
11. State the possible number of imaginary zeros of g(x) � x4 � 3x3 � 7x2 � 6x � 13.A. 3 or 1 B. 2 or 0 C. exactly 1 D. exactly 3 11.
77
xO
f(x )
xO
f(x )
© Glencoe/McGraw-Hill 434 Glencoe Algebra 2
Chapter 7 Test, Form 2B (continued)
12. Write a polynomial function of least degree with integral coefficients whose zeros include �3 and i.A. g(x) � x3 � 3x2 � x � 3 B. g(x) � x3 � 3x2 � x � 3C. g(x) � x2 � 1 D. g(x) � x3 � 3x2 � x � 3 12.
13. List all of the possible rational zeros of p(x) � 2x3 � 6x2 � 7x � 6.
A. �1, �2, �3, �6, ��12�, ��
32� B. �1, �2, �3, �6
C. �1, �2, ��16�, ��
13�, ��
12�, ��
23� D. �1, �2, �3, �6, ��
13�, ��
12�, ��
23� 13.
14. Find all of the rational zeros of g(x) � 2x3 � 11x2 � 8x � 21.
A. �1, 3, �72� B. �1, �3, ��
72� C. �1, 3 D. �1, 3, 7 14.
15. Find (f � g)(x) for f(x) � x2 � 8x and g(x) � 3x � 5.A. �x2 � 5x � 5 B. x2 � 5x � 5 C. x2 � 5x � 5 D. x2 � 11x � 5 15.
16. If f(x) � x2 � 3, and g(x) � 2x � 1, find [g � f ](x).A. 2x3 � x2 � 6x � 3 B. 4x2 � 4x � 2C. x2 � 2x � 4 D. 2x2 � 7 16.
17. State the domain and range of the function graphed at the right.A. D: x � �4, R: y � 0B. D: x �4, R: y 0C. D: x �4, R: y � 0D. D: x � �4, R: y 0 17.
18. Find the inverse of f(x) � 3 � 5x.
A. f �1(x) � 5 � 3x B. f �1(x) � �x �
53
�
C. f �1(x) � �3 �
55x
� D. f �1(x) � �3 � �15�x 18.
19. Determine which pair of functions are not inverse functions.A. g(x) � 2x � 9 B. g(x) � x � 1 C. g(x) � 3x � 6 D. g(x) � 3x � 4 19.
h(x) � �12�x � 9 h(x) � x � 1 h(x) � �
13�x � 2 h(x) � �
x �3
4�
20. Which inequality is graphed at the right?A. y �x � 3� B. y �x � 3�C. y � �x � 3� D. y � �x � 3� 20.
Bonus If g(x) � 4x � 9, solve g [g(x)] � g(x) B:for x.
NAME DATE PERIOD
77
y
xO
xO
y
Chapter 7 Test, Form 2C
© Glencoe/McGraw-Hill 435 Glencoe Algebra 2
1. Find p(�5) if p(x) � x3 � 2x2 � x � 4. 1.
2. Find p(x � 1) if p(x) � x2 � 3x � 1. 2.
3. Determine whether the graph 3.represents an odd-degree or an even-degree polynomial function. Then state the number of real zeros.
4. Graph f(x) � x3 � 3x � 1 by making a table of values. 4.Then determine consecutive values of x between which each real zero is located.
5. For the graph in Question 4, estimate the x-coordinates 5.at which the relative maxima and relative minima occur.
6. Write the expression 9n6 � 36n3 in quadratic form, if 6.possible.
7. Solve x4 � 12x2 � 45 � 0. 7.
8. Use synthetic substitution to find f(�4) for 8.f(x) � x3 � 3x2 � 5x � 7.
9. One factor of x3 � 2x2 � 23x � 60 is x � 4. Find the 9.remaining factors.
10. State the possible number of positive real zeros, 10.negative real zeros, and imaginary zeros for f (x) � 3x4 � 2x3 � 5x2 � 6x � 2.
xO
f(x )
NAME DATE PERIOD
SCORE 77
Ass
essm
ent
xO
f(x )
© Glencoe/McGraw-Hill 436 Glencoe Algebra 2
Chapter 7 Test, Form 2C (continued)
11. Find all the zeros of the function h(x) � x3 � 5x2 � 4x � 20. 11.
12. List all of the possible rational zeros of 12.f(x) � 2x3 � x2 � 4x � 8.
13. Find all of the rational zeros of g(x) � 2x3 � x2 � 7x � 6. 13.
14. Find ( f � g)(x) for f(x) � x2 � 4 and g(x) � 7 � x. 14.
15. If f(x) � x � 5 and g(x) � x2 � 3, find f [g(�2)]. 15.
16. If f(x) � 2x � 5 and g(x) � x2 � 3, find [f � g](x). 16.
17. Find the inverse of f(x) � 5x � 10. 17.
18. Determine whether f(x) � 5x � 3 and g(x) � �x �
53
� are 18.
inverse functions.
19. Graph y � �2x � 8�. Then state the domain and range 19.of the function.
20. Graph y �x � 2�. 20.
Bonus If g(x) � 5x � 8, solve g[ g(x)] � g(x) for x. B:
y
xO
y
xO
NAME DATE PERIOD
77
Chapter 7 Test, Form 2D
© Glencoe/McGraw-Hill 437 Glencoe Algebra 2
1. Find p(�4) if p(x) � x3 � 3x2 � 7x � 6. 1.
2. Find p(x � 1) if p(x) � x2 � 4x � 2. 2.
3. Determine whether the graph represents 3.an odd-degree or an even-degree polynomial function. Then state the number of real zeros.
4. Graph f(x) � �x3 � 3x � 1 by making a table of values. 4.Then determine consecutive values of x between which each real zero is located.
5. For the graph in Question 4, estimate the x-coordinates at 5.which the relative maxima and relative minima occur.
6. Write the expression 5x10 � 4x5 � 3 in quadratic form, if 6.possible.
7. Solve x4 � 4x2 � 12 � 0. 7.
8. Use synthetic substitution to find f(�4) for 8.f(x) � x4 � 7x2 � 12.
9. One factor of g(x) � x3 � x2 � 9x � 9 is x � 3. Find the 9.remaining factors.
10. State the number of positive real zeros, negative real zeros, 10.and imaginary zeros for f(x) � 2x4 � 5x3 � 3x2 � x � 6.
xO
f(x )
NAME DATE PERIOD
SCORE 77
Ass
essm
ent
xO
f(x )
© Glencoe/McGraw-Hill 438 Glencoe Algebra 2
Chapter 7 Test, Form 2D (continued)
11. Find all the zeros of the function p(x) � x3 � 2x2 � 9x � 18. 11.
12. List all of the possible rational zeros of 12.g(x) � 2x3 � 2x2 � 7x � 14.
13. Find all of the rational zeros of h(x) � 3x3 � 4x2 � 13x � 6. 13.
14. Find (f � g)(x) for f(x) � x2 � 4 and g(x) � 6 � x. 14.
15. If f(x) � 2x � 7 and g(x) � x2 � 5, find g[f(5)]. 15.
16. If f(x) � 3 � x and g(x) � x2 � 4, find [ g � f ](x). 16.
17. Find the inverse of g(x) � �2x � 4. 17.
18. Determine whether f(x) � 4x � 8 and g(x) � �14�x � 2 are 18.
inverse functions.
19. Graph y � �3x � 6�. Then state the domain and range of 19.the function.
20. Graph y �2x � 2�. 20.
Bonus If g(x) � 3x � 8, solve g[ g(x)] � g(x) for x. B:
y
xO
y
xO
NAME DATE PERIOD
77
Chapter 7 Test, Form 3
© Glencoe/McGraw-Hill 439 Glencoe Algebra 2
1. Find p(�2) if p(x) � �18�x3 � �
34�x2 � �
12�x � �
43�. 1.
2. If p(x) � 2x2 � 3x � 1 and r(x) � x2 � 5x, find 2.r(x2) � p(x � 1).
3. Describe the end behavior and 3.determine whether the graph represents an odd-degree or an even-degree polynomial function.Then state the number of real zeros.
4. Graph f(x) � �x4 � 3x2 � x � 2 by making a table of 4.values. Then determine the values of x between which the real zeros are located.
5. For the graph in Question 4, estimate the x-coordinates at 5.which the relative maxima and relative minima occur.
6. Write the expression 9b5 � 3b3 � 8b in quadratic form, if 6.possible.
7. Solve x�12�
� 5x�14�
� 6 � 0. 7.
8. Use synthetic substitution to find f(�4) for 8.f(x) � 2x6 � 4x4 � 2x3 � 5x � 6.
9. Find the value of k so that the remainder is 3 for 9.(x2 � x � k) � (x � 1).
10. State the possible number of positive real zeros, negative 10.real zeros, and imaginary zeros for f(x) � 2x10 � 3x8 � 4x6 � x4 � 3x2 � 2.
xO
f(x )
NAME DATE PERIOD
SCORE 77
Ass
essm
ent
xO
f(x )
© Glencoe/McGraw-Hill 440 Glencoe Algebra 2
Chapter 7 Test, Form 3 (continued)
11. Find all of the zeros of the function 11.q(x) � x4 � 8x3 � 22x2 � 8x � 39.
12. List all of the possible rational zeros of 12.h(x) � 9x6 � 12x3 � 15.
13. Find all of the rational zeros of 13.h(x) � 24x4 � 38x3 � 23x2 � 5x � 2.
14. Find (f � g)(x) for f(x) � x2 � 4 and g(x) � �x �x
2�. 14.
15. If g(x) � 3x and h(x) � x3 � x2 � x � 1, find [h � g](x). 15.
16. If f(x) � 5x, g(x) � 2x � 1, and h(x) � x2 � 1, find 16.[h � ( g � f )](�3).
17. Find the inverse of h(x) � �2x
5� 6�. 17.
18. Determine whether f(x) � �12�x � �
73� and g(x) � 2x � �
134� 18.
are inverse functions.
19. Graph y � �x � 4� � 2. Then state the domain and 19.range of the function.
20. Graph y �x � 3� � 3. 20.
Bonus If f(�3) � �120, for f(x) � x4 � x3 � 19x2 � kx � 30, B:find f(1).
y
x
O
y
xO
NAME DATE PERIOD
77
Chapter 7 Open-Ended Assessment
© Glencoe/McGraw-Hill 441 Glencoe Algebra 2
Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solutions in more thanone way or investigate beyond the requirements of the problem.
1. a. Sketch a graph of a polynomial function f(x) of degree 5 thathas the maximum number of real zeros possible for a functionof its degree. Label the zeros z1, z2, … .
b. Label relative maximum points of the graph, if any, A1, A2, …and label relative minimum points of the graph, if any,B1, B2, … .
c. State the domain and range of the function.d. Use the notation “As x → ___ , f(x) → ___” to describe the
end behavior of your graph.
2. The domain of a polynomial function g(x) is all real numbers andthe range of this function is g(x) 2. What do you know aboutthe degree, the leading coefficient, and the zeros of this function?Explain your reasoning.
3. a. Write a fourth-degree polynomial P(x) where no coefficient iszero, that is an 0 for any n.
b. Find P(�2) in two different ways.c. Determine whether x � 1 is a factor of P(x).d. Explain what information Descartes’ Rule of Signs provides
about P(x).e. Explain how to find, then list, all of the possible rational zeros
of P(x).f. Explain how to find, then state, the rational zeros of P(x).
4. a. Write a first-degree function g(x) and a second-degree function h(x).
Find g(2x � 3), h(3a), ( g � h)(x), ( g � h)(x), ( g � h)(x), ��hg��(x),
(h � g)(x), g[h(x)], [h � ( g � g)](2), and g�1(x).b. Explain, then show, how to prove that g(x) and g�1(x) are, in
fact, inverse functions. Then explain the relationship betweenthe graphs of these two functions.
NAME DATE PERIOD
SCORE 77
Ass
essm
ent
© Glencoe/McGraw-Hill 442 Glencoe Algebra 2
Chapter 7 Vocabulary Test/Review
Underline the correct word or phase that best completes each sentence.
1. (End behavior, Composition of functions, Synthetic substitution) is a methodfor evaluating a polynomial function f(x) at a particular value of x.
2. If a function has an inverse that is also a function, then it must be a (one-to-one function, square root function, power function).
3. Writing the polynomial 2x4 � 9x2 � 15 as 2(x2)2 � 9(x2) � 15 uses the ideaof (leading coefficients, quadratic form, end behavior).
4. The (rational zero theorem, remainder theorem, fundamental theorem of algebra)says that every polynomial equation with degree one or greater has at least oneroot in the set of complex numbers.
5. y � �3x � 5� is a(n) (square root, polynomial, inverse) function.
6. If f(x) is a polynomial function such that f(�2) � 8 and f(�3) � �5, then the (depressed polynomial, Location Principle, relative minimum) tells you thatf(x) has at least one real zero between �2 and �3.
7. If a point is on a graph of a polynomial function and no other nearby points of the graph have a lesser y-coordinate, the point is a relative(maximum, minimum) of the function.
8. If x3 � 3x2 � 4x � 12 is divided by x � 2, the quotient will be x2 � 5x � 6and the remainder will be 0. In this case, x2 � 5x � 6 is called the (quadratic form, depressed polynomial, power function).
9. The process of forming a new function from two given functions by performing the two functions in succession is called (synthetic substitution, end behavior, composition of functions).
10. The expression 4x3 � 3x2 � 5x � 6 is a(n) (polynomial function, inverse relation, polynomial in one variable).
In your own words—Define each term.
11. end behavior
12. Factor Theorem
Complex Conjugates Theorem
composition of functionsdegree of a polynomialdepressed polynomialDescartes’ Rule of Signsend behavior
Factor TheoremFundamental Theorem of Algebra
identity functionIntegral Zero Theoreminverse functioninverse relation
leading coefficientLocation Principleone-to-onepolynomial functionpolynomial in one variable
quadratic form
Rational Zero Theoremrelative maximumrelative minimumRemainder Theoremsquare root functionsquare root inequalitysynthetic substitution
NAME DATE PERIOD
SCORE 7777
Chapter 7 Quiz (Lessons 7–1 through 7–3)
77
© Glencoe/McGraw-Hill 443 Glencoe Algebra 2
NAME DATE PERIOD
SCORE
Chapter 7 Quiz (Lessons 7–4 and 7–5)
1. Use synthetic substitution to find f(3) and f(�4) for 1.f(x) � x4 � 8x � 11.
2. One factor of x3 � x2 � 14x � 24 is x � 4. Find the 2.remaining factors.
3. State the possible number of positive real zeros, 3.negative real zeros, and imaginary zeros for g(x) � 3x5 � 2x3 � 4x2 � 8x � 1.
4. Find all of the zeros of f(x) � x3 � 5x2 � 8x � 6. 4.
5. Standardized Test Practice Write a polynomial function of least degree with integral coefficients whose zeros include 4 and 1 � i.A. f(x) � x3 � 2x2 � 6x � 8B. f(x) � x3 � 6x2 � 10x � 8C. f(x) � x3 � 6x2 � 10x � 8D. f(x) � x3 � 6x2 � 10x � 8 5.
NAME DATE PERIOD
SCORE 77
Ass
essm
ent
77
NAME DATE PERIOD
SCORE
1. If p(x) � 3x2 � 2x � 1, find p(�4). 1.
2. Determine whether the graph at the right represents an odd-degree polynomial or an even-degree polynomial function.Then state the number of real zeros.
3. Graph f(x) � x3 � 5x2 � 4x � 3 by making a table of values.Then determine consecutive values of x between which each real zero is located. Estimate the x-coordinates atwhich the relative maxima and relative minima occur.
Solve each equation.
4. x4 � 14x2 � 45 � 0 5. a4 � 49
xO
f(x )2.
3.
4.
5.
xO
f(x )
77
© Glencoe/McGraw-Hill 444 Glencoe Algebra 2
1. List all of the possible rational zeros of 1.h(x) � 2x4 � 5x3 � 3x2 � 4x � 6. Then find all of the rational zeros of the function.
2. Find (f � g)(x), (f � g)(x), (f � g)(x), and ��gf��(x) for 2.
f(x) � x2 � 3x � 2 and g(x) � 2x � 4.
3. For f(x) � {(2, 3), (4, 4), (5, 8)} and g(x) � {(2, 4), (3, 5), (4, 2), 3.(8, 4)}, find f � g and g � f if they exist.
4. Find [g � h](x) and [h � g](x) for g(x) � x2 � 2x � 1 and 4.h(x) � x � 4.
5. If f(x) � 3x � 2 and g(x) � x2 � 1, find f[g(�3)] and g[f(�3)]. 5.
Chapter 7 Quiz (Lessons 7–8 and 7–9)
1. Find the inverse of the relation {(�2, 5), (0, 4), (1, �8), (4, 7)}. 1.
2. Find the inverse of the function f(x) � 4x � 2.Then graph the function and its inverse.
3. Determine whether g(x) � 3x � 6 and f(x) � �13�x � 2 are
inverse functions.
4. Graph y � �3x � 9�. Then state the domain and range of the function.
NAME DATE PERIOD
SCORE
Chapter 7 Quiz (Lessons 7–6 and 7–7)
77
NAME DATE PERIOD
SCORE
77
1.
2.
3.
4.y
xO
xO
f(x )
Chapter 7 Mid-Chapter Test (Lessons 7–1 through 7–5)
© Glencoe/McGraw-Hill 445 Glencoe Algebra 2
Write the letter for the correct answer in the blank at the right of each question.
1. Find p(�4) if p(x) � 3x2 � 4x � 7.A. 7 B. 71 C. 57 D. 39 1.
2. State the degree of 2x2 � 5x3 � 7x4 � 9.A. 4 B. 7 C. �9 D. 3 2.
For Questions 3 and 4, use the graph shown.
3. State the number of real zeros of the function.A. 2 B. 4C. 1 D. 3 3.
4. As x → � �, f(x) →�?��
describes the end behavior of the graph.A. �∞ B. 0 C. �∞ D. x 4.
5. Use synthetic substitution to find f(�2) for f(x) � x3 � 6x2 � 5x � 1.A. �41 B. 7 C. �21 D. 27 5.
6. One factor of x3 � 6x2 � x � 6 is x � 6. Find the remaining factors.A. x � 1, x � 1 B. x, x � 1 C. x � 1, x � 1 D. x � 1, x � 1 6.
7. Graph f(x) � x3 � 4x2 � 5 by using a table of values. 7.Then determine consecutive values of x between which each real zero is located.
8. Solve t5 � 81t � 0. 8.
9. State the possible number of positive real zeros, 9.negative real zeros, and imaginary zeros of f(x) � x4 � 3x3 � 2x2 � x � 1.
10. Write a polynomial function of least degree with integral 10.coefficients whose zeros include 3 and 3i.
11. Find all the zeros of the function f(x) � x3 � x2 � 16x � 16. 11.
xO
f(x )
Part I
NAME DATE PERIOD
SCORE 77
Ass
essm
ent
xO
f(x )
Part II
© Glencoe/McGraw-Hill 446 Glencoe Algebra 2
Chapter 7 Cumulative Review (Chapters 1–7)
1. Define a variable and write an inequality. Then solve. 1.Marlea received an inheritance of $10,000. She plans to invest some in a stock that pays 7% interest annually. She will deposit the remainder in a savings account that pays 5% interest annually. What is the least amount that Marlea can invest in stock if she wants to earn at least $550 on her investments for the year? (Lesson 1-5)
2. Describe the system of equations as consistent and 2.independent, consistent and dependent, or inconsistent.6x � 2y � 49x � 3y � 6 (Lesson 3-1)
3. Triangle ABC with vertices at A(�1, �3), B(2, 3), and 3.C(�4, 1) is translated 5 units right and 3 units down.Find the coordinates of A�, B�, and C�. (Lesson 4-4)
4. Use Cramer’s Rule to solve the system of equations. 4.2x � y � �1�3x � y � 4 (Lesson 4-6)
5. Factor 4n2 � 20n � 25 completely. If the polynomial is not 5.factorable, write prime. (Lesson 5-4)
6. Solve �2x � 1�0� � 1 � 5. (Lesson 5-8) 6.
7. Graph the quadratic function f(x) � x2 � 2x � 8, labeling 7.the y-intercept, vertex, and axis of symmetry. (Lesson 6-1)
8. Write a quadratic equation with 3 and �2 as its roots.Write the equation in the form ax2 � bx � c � 0, where a, b, and c are integers. (Lesson 6-3) 8.
9. Find p(�3) if p(x) � x4 � 8x3 � 5x � 4. (Lesson 7-1) 9.
10. State the number of possible positive real zeros, 10.negative real zeros, and imaginary zeros for h(x) � x4 � 2x3 � 5x � 4. (Lesson 7-5)
11. Find all of the rational zeros of f(x) � 2x3 � x2 � 13x � 6. 11.(Lesson 7-6)
12. Find the inverse of f(x) � 5x � 4. (Lesson 7-8) 12.
NAME DATE PERIOD
77
xOf(x )
Standardized Test Practice (Chapters 1–7)
© Glencoe/McGraw-Hill 447 Glencoe Algebra 2
1. If r2 � 1 � �2r, then �r � �12��2
� .
A. ��14� B. �
14�
C. 1 D. cannot be determined 1.
2. How many fourths is 26�23�%?
E. �14� F. 4 G. �
1165�
H. �145�
2.
3. Find p in terms of m if �mp� � q, q � p, p 0, and m 0.
A. ��m� B. ��mq� C. m D. ��p� 3.
4. Find the average of �a3�, �
a6�, and �
a9�.
E. �1514a
� F. �1118a
� G. �5a4�
H. �161a� 4.
5. What is the ones digit in 350?A. 1 B. 3 C. 7 D. 9 5.
6. What is the value of a2 � b2 if a � b � 6 and a � b � �3?E. �18 F. 3 G. 9 H. 18 6.
7. If �n� is an irrational number, which of the following must be irrational?
A. �n2� B. 2�n� C. ��n2�� D. �2n� 7.
8. Evaluate 4m3 � 3m2 � 2m � 2 if m � �1.E. 1 F. �11 G. �1 H. 7 8.
9. If the slope of the line through A(�7, 4) and B(5, y) is ��14�, what is
the value of y?
A. �1 B. 7 C. �92� D. 1 9.
10. Find the area of square ABCD.
E. �53� units2 F. 25 units2
G. 625 units2 H. �295� units2 10. HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
?
NAME DATE PERIOD
77
Ass
essm
ent
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
9x2
30x � 25
B C
A D
© Glencoe/McGraw-Hill 448 Glencoe Algebra 2
Standardized Test Practice (continued)
11. Of the 24 socks in a drawer, there are three 11. 12.times as many black socks as brown socks.Some of the black socks are plain and some are patterned. There are five times as many plain socks as there are patterned socks.What is the probability that, without looking,you select a plain black sock from the drawer?
12. Let d� and d* be defined for any positive integer d as follows: d� is the number obtained by dividing d by its first digit and d* is the sum of the digits of d. What is
the value of �335544�*�?
13. Point X lies between points P and Q on a 13. 14.number line. If XQ � 15 and PQ � 24,then PX � .
14. If �18� � �0
n.4�
, what is the value of n?
Column A Column B
15. 15.
16. 16.
17. 17. DCBA(x � 1)2(x � 2)2
DCBA8 � 6 � (9 � 3)8 � 6 � 9 � 3
DCBAthe number of
faces of a cubethe number of
vertices of a pentagon
Part 3: Quantitative Comparison
Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.
?0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
NAME DATE PERIOD
77
NAME DATE PERIOD
A
D
C
B
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
Unit 2 Test (Chapters 5–7)
© Glencoe/McGraw-Hill 449 Glencoe Algebra 2
For Questions 1–7, simplify. Assume that no denominator equals 0.
1. (7x2 � 3x � 9) � (�x2 � 8x � 3)
2. 5x3(7x)2 3. (2x � 3)2
4. 5. �16x2y�4�
6. �12� � �18� � 3�50� � �75� 6.
7. �12
��
3ii� 7.
8. Use synthetic division to find (2x3 � 5x2 � 7x � 1) � (x � 1). 8.
9. Write the expression m�79�
in radical form. 9.
10. Solve �3x � 6� � 4 � 7. 10.
11. Graph f(x) � �x2 � 4x � 3, labeling the y-intercept, vertex, 11.and axis of symmetry.
12. The shape of a supporting arch can be modeled by h(x) � �0.03x2 � 3x, where h(x) represents the height of the 12.arch and x represents the horizontal distance from one end of the base of the arch in meters. Find the maximum height of the arch.
13. Solve 2x2 � 3x � 2 by graphing. If exact roots cannot be 13.found, state the consecutive integers between which the roots are located.
14. Solve x2 � 2x � 24 by factoring. 14.
15. Write a quadratic equation with ��34
� and 4 as its roots. 15.
Write the equation in the form ax2 � bx � c � 0, where a, b, and c are integers.
16. Find the exact solutions to 6x2 � x � 4 � 0 by using the 16.Quadratic Formula.
17. Find the value of the discriminant for 9x2 � 1 � 6x. Then 17.describe the number and type of roots for the equation.
xO
f(x )
8y3 � 27���2xy � 10y � 3x � 15
NAME DATE PERIOD
SCORE
Ass
essm
ent
1.
2.
3.
4.
5.
© Glencoe/McGraw-Hill 450 Glencoe Algebra 2
Unit 2 Test (continued)(Chapters 5–7)
18. Identify the vertex, axis of symmetry, and direction of opening 18.for y � 2(x � 3)2 � 5.
19. Write y � �4x2 � 8x � 1 in vertex form. 19.
20. Graph y � x2 � 2x � 1. 20.
21. Find p(�3) if p(x) � x5 � 3x2. 21.
22. Graph f(x) � (�x)4 � 4x2 � 2x by making a table of values. 22.Then estimate the x-coordinates at which the relative maxima and relative minima occur.
23. Solve �x4 � 200 � 102x2. 23.
24. Use synthetic substitution to find f(�3) for 24.f(x) � 2x3 � 6x2 � 5x � 7.
25. One factor of f(x) � x3 � x2 � 22x � 40 is x � 4. Find the 25.other factors.
26. State the number of positive real zeros, negative real zeros, 26.and imaginary zeros for g(x) � 9x3 � 7x2 � 10x � 4.
27. List all of the possible rational zeros of 27.f(x) � 3x5 � 7x3 � 2x � 15.
28. If f(x) � 3x and g(x) � 4x � 3, find f [g(5)] and g[ f(5)]. 28.
29. Find the inverse of f(x) � 7x � 2. 29.
30. Graph y �3x � 1�2�. 30.y
xO
xO
f(x )
y
xO
NAME DATE PERIOD
First Semester Test (Chapters 1–7)
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 451 Glencoe Algebra 2
For Questions 1–18, write the letter for the correct answer in the blank at the right of each question.
1. Name the sets of numbers to which �14� belongs.A. rational numbers B. irrational numbersC. rational numbers, real numbers D. irrational numbers, real numbers 1.
2. Find the range of the relation {(�3, 0), (�2, 0), (1, 4)}. Then determine whether the relation is a function.A. {0, 4}; function B. {0, 4}; not a functionC. {�3, �2, 1}; function D. {�3, �2, 1}; not a function 2.
3. The graph of which equation is a line with undefined slope that passes through (5, 1)?A. y � 1 B. y � 5 C. x � 1 D. x � 5 3.
4. Which point does not satisfy the inequality y � 2x � 3 �?A. (0, 2) B. (�1, �3) C. (1, 3) D. (2, 0) 4.
5. To solve the system of equations 3x � y � 5 and 2x � 3y � 18, which expression could be substituted for y into the second equation?
A. 5 � 3x B. 3x � 5 C. 6 � �23�x D. 18 � 2x 5.
6. Which system of inequalities is graphed?A. y 2x � 4 B. y 2x � 4
y � ��23�x � 2 y ��
23�x � 2
C. y � 2x � 4 D. y � 2x � 4 6.
y � ��23�x � 2 y ��
23�x � 2
7. Find the maximum value of the function f(x, y) � 2x � 3y for the feasible region shown.A. 14 B. 6C. 11 D. �6 7.
8. Which statement is not true for the matrices
A � � , B � � , C � � , and the constant c � �2?
A. A(B � C) � (B � C)A B. A(B � C) � A(C � B)C. c(A � B) � (A � B)c D. A(B � C) � AB � AC 8.
7 59 1
4 7�6 3
2 �30 1
y
xO
y
xO
(1, 4)
(3, 0)(�3, 0)A
sses
smen
t
© Glencoe/McGraw-Hill 452 Glencoe Algebra 2
First Semester Test (continued)(Chapters 1–7)
9. Find the value of .A. 58 B. 47 C. 23 D. �1 9.
10. Cramer’s Rule is used to solve the system of 3x � y � 2z � 1equations at the right. Which determinant 4x � 2y � z � �2represents the numerator for y? 2x � 3y � 3z � 4
A. B. C. D. 10.
11. Evaluate �260
��
1100�
�
2
5�. Express the result in scientific notation.
A. 0.3 � 10 B. 3 � 102 C. 0.3 � 10�3 D. 3 � 10�4 11.
12. Simplify (5 � 2�3�)(2 � 4�3�).A. 10 � 8�3� B. �62 � 16�3� C. �14 D. �14 � 16�3� 12.
13. Solve �3
y � 3� � 6 � 4.A. 1003 B. 103 C. �5 D. 11 13.
14. The quadratic equation 9x2 � 6x � 1 � 9 is to be solved by completing the square. Which equation would not be a step in that solution?
A. �x � �13��2
� 1 B. x � ��13� � 1
C. 9x2 � 6x � 8 � 0 D. x2 � �23�x � �
19� � 1 14.
15. Solve the inequality �x2 � 25 0.A. {x � x �5 or x � 5} B. {x � x � �5 or x � 5}C. {x � �5 x 5} D. � 15.
16. Write the expression 2n�23�
� 3n�13�
� 5 in quadratic form, if possible.
A. �2n�13��
2
� 3�n�13�� � 5 B. 2�n�
13��
2
� 3�n�13�� � 5
C. 2(n2)�13�
� 3(n)�13�
� 5 D. not possible 16.
17. Find all of the rational zeros of f(x) � x5 � 2x3 � 24x.A. 0, �2, � �6� B. 0, �2 C. �6, 0, 4D. 0, �1, �2, �3, �4, �6, �8, �12, �24 17.
18. Determine which pair of functions are not inverses.
A. g(x) � 3x � 5 B. g(x) � �43�x C. g(x) � 2x � 8 D. g(x) � �3
x� � 1 18.
h(x) � �x �
35
� h(x) � �34�x h(x) � �2
x� � 8 h(x) � 3x � 3
3 2 14 1 �22 3 4
3 �1 14 2 �22 3 4
3 1 24 �2 12 4 3
3 �1 24 2 12 3 3
7 112 5
NAME DATE PERIOD
First Semester Test (continued)(Chapters 1–7)
© Glencoe/McGraw-Hill 453 Glencoe Algebra 2
19. The formula for the area A of a circle with diameter d is 19.
A � ���d2��2
. Find the area of a circle with a diameter of
40 centimeters. Use 3.14 for �.
20. Simplify 7(4 � x) � 5(x � 1). 20.
21. Define a variable and write an inequality. Then solve. A 21.shelf in a lumber yard will safely hold up to 1000 pounds.A crate on the shelf is marked 270 pounds. What is the greatest number of sheets of plywood, each weighing 7 pounds, that may safely be stacked on the shelf?
22. Solve 10 � 9 � 2(1 � m) 19. Describe the solution set 22.using set builder or interval notation. Then graph the solution set on a number line.
23. If f(x) � 4(x � 3) � x2, find f(2a). 23.
24. What is the slope of a line that is perpendicular to the 24.graph of the line passing through (2, 1) and (3, 5)?
25. Graph the piecewise function f(x) � � 3x if x � 1 25.
Identify the domain and range. �2x � 2 if x � 1.
26. Solve the system of equations 7x � 3y � �1 and 2x � y � 9 by using elimination.
At a school-sponsored car wash, the fees charged were:$5 per car, $8 per pickup truck, $10 per full-size van. 26.Twice as many cars were washed as pickup trucks. The amount collected for washing cars and pickup trucks was $360. A total of $410 was collected at the car wash.
27. Let c represent the number of cars washed, t represent the 27.number of pickup trucks washed, and v represent the number of vans washed. Write a system of three equations that represents the number of vehicles washed.
28. Find the number of cars washed. 28.
For Questions 29 and 30, perform the indicated matrix operations. If the matrix does not exist, write impossible. 29.
29. �4� � 3� 30. � � � 30.2 0
�3 �41 5
12 4 �9�5 0 3
7 �41 4
6 �42 3
xO
f(x )
3 40 1 2�1
NAME DATE PERIOD
Ass
essm
ent
© Glencoe/McGraw-Hill 454 Glencoe Algebra 2
First Semester Test (continued)(Chapters 1–7)
31. Evaluate using diagonals. 31.
32. Find the inverse of A � � , if it exists. 32.
For Questions 33–35, simplify. Assume that no variable 33.equals 0.
33. (3x2)2(4y5)(2x0y�2) 34. (2x � 3y)(x � 5y) 34.
35. 2i(3 � 4i) � (�1 � i) 35.
36. Use a calculator to approximate �3
287� to three decimal 36.places.
37. Determine whether the function f(x) � 3x2 � 6x � 11 has a 37.minimum or a maximum value and find that value.
38. Solve the equation x2 � 2x � 63 � 0 by factoring. 38.
39. Solve the equation x2 � 4x � 13 � 0 by completing the 39.square.
40. Find the exact solutions to 7x2 � 6x � 1 by using the 40.Quadratic Formula.
41. Write an equation for the parabola with vertex at (�2, 3) 41.and y-intercept �1.
42. One factor of x3 � 5x2 � 8x � 12 is x � 2. Find the 42.remaining factors.
43. Find all the zeros of the function f(x) � x3 � 6x2 � 16x � 96. 43.
44. Find (f � g)(x) and (f � g)(x) for f(x) � 4x � 9 and g(x) � 3x2. 44.
45. Find the inverse of the function p(x) � 4x � 8. 45.
46. Graph y � ��2x � 5�. Then state the domain and range of 46.the function. y
xO
8 2�2 1
3 �4 51 �1 �20 2 3
NAME DATE PERIOD
Standardized Test PracticeStudent Record Sheet (Use with pages 406–407 of the Student Edition.)
© Glencoe/McGraw-Hill A1 Glencoe Algebra 2
NAME DATE PERIOD
77
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7 10
2 5 8 11
3 6 9 12
Solve the problem and write your answer in the blank.
Also enter your answer by writing each number or symbol in a box. Then fill inthe corresponding oval for that number or symbol.
13 15 17 19
14 16 18
Select the best answer from the choices given and fill in the corresponding oval.
20 22 24
21 23 DCBADCBA
DCBADCBADCBA
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
DCBADCBADCBADCBA
DCBADCBADCBADCBA
DCBADCBADCBADCBA
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 3 Quantitative ComparisonPart 3 Quantitative Comparison
© Glencoe/McGraw-Hill A2 Glencoe Algebra 2
Answers (Lesson 7-1)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Po
lyn
om
ial F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-1
7-1
©G
lenc
oe/M
cGra
w-H
ill37
5G
lenc
oe A
lgeb
ra 2
Lesson 7-1
Poly
no
mia
l Fu
nct
ion
s
Apo
lyno
mia
l of
degr
ee n
in o
ne v
aria
ble
xis
an
expr
essi
on o
f th
e fo
rm
Po
lyn
om
ial i
na 0
xn
�a 1
xn
�1
�…
�a n
�2x
2�
a n�
1x�
a n,
On
e V
aria
ble
whe
re t
he c
oeffi
cien
ts a
0, a
1, a
2, …
, a n
repr
esen
t re
al n
umbe
rs,
a 0is
not
zer
o,
and
nre
pres
ents
a n
onne
gativ
e in
tege
r.
Th
e d
egre
e of
a p
olyn
omia
lin
on
e va
riab
le i
s th
e gr
eate
st e
xpon
ent
of i
ts v
aria
ble.
Th
ele
adin
g co
effi
cien
tis
th
e co
effi
cien
t of
th
e te
rm w
ith
th
e h
igh
est
degr
ee.
Apo
lyno
mia
l fun
ctio
n of
deg
ree
nca
n be
des
crib
ed b
y an
equ
atio
n of
the
for
m
Po
lyn
om
ial
P(x
) �
a 0x
n�
a 1x
n�
1�
… �
a n�
2x2
�a n
�1x
�a n
,F
un
ctio
nw
here
the
coe
ffici
ents
a0,
a1,
a2,
…,
a nre
pres
ent
real
num
bers
, a 0
is n
ot z
ero,
an
d n
repr
esen
ts a
non
nega
tive
inte
ger.
Wh
at a
re t
he
deg
ree
and
lea
din
g co
effi
cien
t of
3x2
�2x
4�
7 �
x3?
Rew
rite
th
e ex
pres
sion
so
the
pow
ers
of x
are
in d
ecre
asin
g or
der.
�2x
4�
x3�
3x2
�7
Th
is i
s a
poly
nom
ial
in o
ne
vari
able
.Th
e de
gree
is
4,an
d th
e le
adin
g co
effi
cien
t is
�2.
Fin
d f
(�5)
if
f(x)
�x3
�2x
2�
10x
�20
.f(
x) �
x3�
2x2
�10
x�
20O
rigin
al f
unct
ion
f(�
5) �
(�5)
3�
2(�
5)2
�10
(�5)
�20
Rep
lace
xw
ith �
5.
��
125
�50
�50
�20
Eva
luat
e.
��
5S
impl
ify.
Fin
d g
(a2
�1)
if
g(x)
�x2
�3x
�4.
g(x)
�x2
�3x
�4
Orig
inal
fun
ctio
n
g(a2
�1)
�(a
2�
1)2
�3(
a2�
1) �
4R
epla
ce x
with
a2
�1.
�a4
�2a
2�
1 �
3a2
�3
�4
Eva
luat
e.
�a4
�a2
�6
Sim
plify
.
Sta
te t
he
deg
ree
and
lea
din
g co
effi
cien
t of
eac
h p
olyn
omia
l in
on
e va
riab
le.I
f it
is
not
a p
olyn
omia
l in
on
e va
riab
le,e
xpla
in w
hy.
8;8
1.3x
4�
6x3
�x2
�12
4;3
2.10
0 �
5x3
�10
x77;
103.
4x6
�6x
4�
8x8
�10
x2�
20
4.4x
2�
3xy
�16
y25.
8x3
�9x
5�
4x2
�36
6.�
��
no
t a
po
lyn
om
ial i
n
5;�
9o
ne
vari
able
;co
nta
ins
6;�
two
var
iab
les
Fin
d f
(2)
and
f(�
5) f
or e
ach
fu
nct
ion
.
7.f(
x) �
x2�
98.
f(x)
�4x
3�
3x2
�2x
�1
9.f(
x) �
9x3
�4x
2�
5x�
7�
5;16
23;
�58
673
;�
1243
1 � 25
1 � 72x3� 36
x6� 25
x2� 18
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exam
ple3
Exam
ple3
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill37
6G
lenc
oe A
lgeb
ra 2
Gra
ph
s o
f Po
lyn
om
ial F
un
ctio
ns
If th
e de
gree
is e
ven
and
the
lead
ing
coef
ficie
nt is
pos
itive
, th
enf(
x) →
��
as x
→�
�
f(x)
→�
�as
x→
��
If th
e de
gree
is e
ven
and
the
lead
ing
coef
ficie
nt is
neg
ativ
e, t
hen
En
d B
ehav
ior
f(x)
→�
�as
x→
��
of
Po
lyn
om
ial
f(x)
→�
�as
x→
��
Fu
nct
ion
sIf
the
degr
ee is
odd
and
the
lead
ing
coef
ficie
nt is
pos
itive
, th
enf(
x) →
��
as x
→�
�
f(x)
→�
�as
x→
��
If th
e de
gree
is o
dd a
nd t
he le
adin
g co
effic
ient
is n
egat
ive,
the
nf(
x) →
��
as x
→�
�
f(x)
→�
�as
x→
��
Rea
l Zer
os
of
The
max
imum
num
ber
of z
eros
of
a po
lyno
mia
l fun
ctio
n is
equ
al t
o th
e de
gree
of
the
poly
nom
ial.
a P
oly
no
mia
lA
zero
of
a fu
nctio
n is
a p
oint
at
whi
ch t
he g
raph
inte
rsec
ts t
he x
-axi
s.
Fu
nct
ion
On
a gr
aph,
cou
nt t
he n
umbe
r of
rea
l zer
os o
f th
e fu
nctio
n by
cou
ntin
g th
e nu
mbe
r of
tim
es t
hegr
aph
cros
ses
or t
ouch
es t
he x
-axi
s.
Det
erm
ine
wh
eth
er t
he
grap
h r
epre
sen
ts a
n o
dd
-deg
ree
pol
ynom
ial
or a
n e
ven
-deg
ree
pol
ynom
ial.
Th
en s
tate
th
e n
um
ber
of
real
zer
os.
As
x→
��
,f(x
) →
��
and
as x
→�
�,f
(x)
→�
�,
so i
t is
an
odd
-deg
ree
poly
nom
ial
fun
ctio
n.
Th
e gr
aph
in
ters
ects
th
e x-
axis
at
1 po
int,
so t
he
fun
ctio
n h
as 1
rea
l ze
ro.
Det
erm
ine
wh
eth
er e
ach
gra
ph
rep
rese
nts
an
od
d-d
egre
e p
olyn
omia
l or
an
eve
n-
deg
ree
pol
ynom
ial.
Th
en s
tate
th
e n
um
ber
of
real
zer
os.
1.2.
3.
even
;6
even
;1
do
ub
le z
ero
od
d;
3
x
f (x)
Ox
f (x)
Ox
f (x)
O
x
f (x)
O
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Po
lyn
om
ial F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-1
7-1
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A3 Glencoe Algebra 2
An
swer
s
Answers (Lesson 7-1)
Skil
ls P
ract
ice
Po
lyn
om
ial F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-1
7-1
©G
lenc
oe/M
cGra
w-H
ill37
7G
lenc
oe A
lgeb
ra 2
Lesson 7-1
Sta
te t
he
deg
ree
and
lea
din
g co
effi
cien
t of
eac
h p
olyn
omia
l in
on
e va
riab
le.I
f it
is
not
a p
olyn
omia
l in
on
e va
riab
le,e
xpla
in w
hy.
1.a
�8
1;1
2.(2
x�
1)(4
x2�
3)3;
8
3.�
5x5
�3x
3�
85;
�5
4.18
�3y
�5y
2�
y5�
7y6
6;7
5.u
3�
4u2 v
2�
v46.
2r�
r2�
No
,th
is p
oly
no
mia
lco
nta
ins
two
N
o,t
his
is n
ot
a p
oly
no
mia
l bec
ause
vari
able
s,u
and
v.
� r1 2�ca
nn
ot
be
wri
tten
in t
he
form
rn,
wh
ere
nis
a n
on
neg
ativ
e in
teg
er.
Fin
d p
(�1)
an
d p
(2)
for
each
fu
nct
ion
.
7.p(
x) �
4 �
3x7;
�2
8.p(
x) �
3x�
x2�
2;10
9.p(
x) �
2x2
�4x
�1
7;1
10.p
(x)
��
2x3
�5x
�3
0;�
3
11.p
(x)
�x4
�8x
2�
10�
1;38
12.p
(x)
��1 3� x
2�
�2 3� x�
23;
2
If p
(x)
�4x
2�
3 an
d r
(x)
�1
�3x
,fin
d e
ach
val
ue.
13.p
(a)
4a2
�3
14.r
(2a)
1 �
6a
15.3
r(a)
3 �
9a16
.�4p
(a)
�16
a2�
12
17.p
(a2 )
4a4
�3
18.r
(x�
2)7
�3x
For
eac
h g
rap
h,
a.d
escr
ibe
the
end
beh
avio
r,b
.det
erm
ine
wh
eth
er i
t re
pre
sen
ts a
n o
dd
-deg
ree
or a
n e
ven
-deg
ree
pol
ynom
ial
fun
ctio
n,a
nd
c.st
ate
the
nu
mb
er o
f re
al z
eroe
s.
19.
20.
21.
f(x
) →
��
as x
→�
�,
f(x
) →
��
as x
→�
�,
f(x
) →
��
as x
→�
�,
f(x
) →
��
as x
→�
�;
f(x
) →
��
as x
→�
�;
f(x
) →
��
as x
→�
�;
od
d;
1ev
en;
4o
dd
;3
x
f (x)
Ox
f (x)
Ox
f (x)
O
1 � r2
©G
lenc
oe/M
cGra
w-H
ill37
8G
lenc
oe A
lgeb
ra 2
Sta
te t
he
deg
ree
and
lea
din
g co
effi
cien
t of
eac
h p
olyn
omia
l in
on
e va
riab
le.I
f it
is
not
a p
olyn
omia
l in
on
e va
riab
le,e
xpla
in w
hy.
1.(3
x2�
1)(2
x2�
9)4;
62.
�1 5� a3
��3 5� a
2�
�4 5� a3;
�1 5�
3.�
3m�
12N
ot
a p
oly
no
mia
l;4.
27 �
3xy3
�12
x2y2
�10
y
� m2 2�ca
nn
ot
be
wri
tten
in t
he
form
No
,th
is p
oly
no
mia
l co
nta
ins
two
mn
for
a n
on
neg
ativ
e in
teg
er n
.va
riab
les,
xan
d y
.
Fin
d p
(�2)
an
d p
(3)
for
each
fu
nct
ion
.
5.p(
x) �
x3�
x56.
p(x)
��
7x2
�5x
�9
7.p(
x) �
�x5
�4x
3
24;
�21
6�
29;
�39
0;�
135
8.p(
x) �
3x3
�x2
�2x
�5
9.p(
x) �
x4�
�1 2� x3
��1 2� x
10.p
(x)
��1 3� x
3�
�2 3� x2
�3x
�37
;73
13;
93�
6;24
If p
(x)
�3x
2�
4 an
d r
(x)
�2x
2�
5x�
1,fi
nd
eac
h v
alu
e.
11.p
(8a)
12.r
(a2 )
13.�
5r(2
a)
192a
2�
42a
4�
5a2
�1
�40
a2�
50a
�5
14.r
(x�
2)15
.p(x
2�
1)16
.5[p
(x�
2)]
2x2
�3x
�1
3x4
�6x
2�
115
x2�
60x
�40
For
eac
h g
rap
h,
a.d
escr
ibe
the
end
beh
avio
r,b
.det
erm
ine
wh
eth
er i
t re
pre
sen
ts a
n o
dd
-deg
ree
or a
n e
ven
-deg
ree
pol
ynom
ial
fun
ctio
n,a
nd
c.st
ate
the
nu
mb
er o
f re
al z
eroe
s.
17.
18.
19.
f(x
) →
��
as x
→�
�,
f(x
) →
��
as x
→�
�,
f(x
) →
��
as x
→�
�,
f(x
) →
��
as x
→�
�;
f(x
) →
��
as x
→�
�;
f(x
) →
��
as x
→�
�;
even
;2
even
;1
od
d;
5
20.W
IND
CH
ILL
Th
e fu
nct
ion
C(s
) �
0.01
3s2
�s
�7
esti
mat
es t
he
win
d ch
ill
tem
pera
ture
C(s
) at
0�F
for
win
d sp
eeds
sfr
om 5
to
30 m
iles
per
hou
r.E
stim
ate
the
win
d ch
ill
tem
pera
ture
at
0�F
if
the
win
d sp
eed
is 2
0 m
iles
per
hou
r.ab
ou
t �
22�F
x
f (x)
Ox
f (x)
Ox
f (x)
O
2� m
2
Pra
ctic
e (
Ave
rag
e)
Po
lyn
om
ial F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-1
7-1
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 7-1)
Readin
g t
o L
earn
Math
em
ati
csP
oly
no
mia
l Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-1
7-1
©G
lenc
oe/M
cGra
w-H
ill37
9G
lenc
oe A
lgeb
ra 2
Lesson 7-1
Pre-
Act
ivit
yW
her
e ar
e p
olyn
omia
l fu
nct
ion
s fo
un
d i
n n
atu
re?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-1
at
the
top
of p
age
346
in y
our
text
book
.
•In
the
hon
eyco
mb
cros
s se
ctio
n sh
own
in y
our
text
book
,the
re i
s 1
hexa
gon
in t
he
cen
ter,
6 h
exag
ons
in t
he
seco
nd
rin
g,an
d 12
hex
agon
s in
th
e th
ird
ring
.How
man
y he
xago
ns w
ill t
here
be
in t
he f
ourt
h,fi
fth,
and
sixt
h ri
ngs?
18;
24;
30•
Th
ere
is 1
hex
agon
in
a h
oney
com
b w
ith
1 r
ing.
Th
ere
are
7 h
exag
ons
ina
hon
eyco
mb
wit
h 2
rin
gs.H
ow m
any
hex
agon
s ar
e th
ere
in h
oney
com
bsw
ith
3 r
ings
,4 r
ings
,5 r
ings
,an
d 6
rin
gs?
19;
37;
61;
91
Rea
din
g t
he
Less
on
1.G
ive
the
degr
ee a
nd
lead
ing
coef
fici
ent
of e
ach
pol
ynom
ial
in o
ne
vari
able
.
deg
ree
lead
ing
coef
fici
ent
a.10
x3�
3x2
�x
�7
b.
7y2
�2y
5�
y�
4y3
c.10
0
2.M
atch
eac
h d
escr
ipti
on o
f a
poly
nom
ial
fun
ctio
n f
rom
th
e li
st o
n t
he
left
wit
h t
he
corr
espo
ndi
ng
end
beh
avio
r fr
om t
he
list
on
th
e ri
ght.
a.ev
en d
egre
e,n
egat
ive
lead
ing
coef
fici
ent
iiii.
f(x)
→�
�as
x→
��
;f(
x) →
��
as x
→�
�
b.
odd
degr
ee,p
osit
ive
lead
ing
coef
fici
ent
ivii
.f(x
) →
��
as x
→�
�;
f(x)
→�
�as
x→
��
c.od
d de
gree
,neg
ativ
e le
adin
g co
effi
cien
tii
iii.
f(x)
→�
�as
x→
��
;f(
x) →
��
as x
→�
�
d.
even
deg
ree,
posi
tive
lea
din
g co
effi
cien
ti
iv.
f(x)
→�
�as
x→
��
;f(
x) →
��
as x
→�
�
Hel
pin
g Y
ou
Rem
emb
er
3.W
hat
is
an e
asy
way
to
rem
embe
r th
e di
ffer
ence
bet
wee
n t
he
end
beh
avio
r of
th
e gr
aph
sof
eve
n-d
egre
e an
d od
d-de
gree
pol
ynom
ial
fun
ctio
ns?
Sam
ple
an
swer
:B
oth
en
ds
of
the
gra
ph
of
an e
ven
-deg
ree
fun
ctio
nev
entu
ally
kee
p g
oin
g in
th
e sa
me
dir
ecti
on
.Fo
r o
dd
-deg
ree
fun
ctio
ns,
the
two
en
ds
even
tual
ly h
ead
in o
pp
osi
te d
irec
tio
ns,
on
e u
pw
ard
,th
eo
ther
do
wn
war
d.
100
0
�2
510
3
©G
lenc
oe/M
cGra
w-H
ill38
0G
lenc
oe A
lgeb
ra 2
Ap
pro
xim
atio
n b
y M
ean
s o
f P
oly
no
mia
lsM
any
scie
nti
fic
expe
rim
ents
pro
duce
pai
rs o
f n
um
bers
[x,
f(x)
] th
at c
an
be r
elat
ed b
y a
form
ula
.If
the
pair
s fo
rm a
fu
nct
ion
,you
can
fit
a
poly
nom
ial
to t
he
pair
s in
exa
ctly
on
e w
ay.C
onsi
der
the
pair
s gi
ven
by
the
foll
owin
g ta
ble.
We
wil
l as
sum
e th
e po
lyno
mia
l is
of
degr
ee t
hree
.Sub
stit
ute
the
give
n va
lues
int
o th
is e
xpre
ssio
n.
f(x)
�A
�B
(x�
x 0)
�C
(x�
x 0)(
x�
x 1)
�D
(x�
x 0)(
x�
x 1)(
x�
x 2)
You
wil
l get
the
sys
tem
of
equa
tion
s sh
own
belo
w.Y
ou c
an s
olve
thi
s sy
stem
an
d us
e th
e va
lues
for
A,B
,C,a
nd D
to f
ind
the
desi
red
poly
nom
ial.
6 �
A11
�A
�B
(2 �
1) �
A�
B39
�A
�B
(4 �
1) �
C(4
�1)
(4 �
2) �
A�
3B�
6C�
54 �
A�
B(7
�1)
�C
(7 �
1)(7
�2)
�D
(7 �
1)(7
�2)
(7 �
4) �
A�
6B�
30C
�90
D
Sol
ve.
1.S
olve
the
sys
tem
of
equa
tion
s fo
r th
e va
lues
A,B
,C,a
nd D
.A
�6,
B�
5,C
�3,
D�
�2
2.F
ind
the
poly
nom
ial
that
rep
rese
nts
the
four
ord
ered
pai
rs.W
rite
you
r an
swer
in
the
form
y�
a�
bx�
cx2
�d
x3.
y�
�2x
3�
17x
2�
32x
�23
3.F
ind
the
poly
nom
ial
that
giv
es t
he f
ollo
win
g va
lues
.
A�
�20
7,B
�94
,C�
25,D
�1;
y�
x3�
10x2
�10
x�
1
4.A
sci
enti
st m
easu
red
the
volu
me
f(x)
of
carb
on d
ioxi
de g
as t
hat
can
be
abso
rbed
by
one
cubi
c ce
ntim
eter
of
char
coal
at
pres
sure
x.F
ind
the
valu
es f
or A
,B,C
,and
D.
A�
3.1,
B�
0.01
091,
C�
�0.
0000
0643
,D�
0.00
0000
0066
x12
034
053
469
8
f(x
)3.
15.
57.
18.
3
x8
1215
20
f(x
)�
207
169
976
3801
x1
24
7
f(x
)6
1139
�54
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-1
7-1
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
An
swer
s
Answers (Lesson 7-2)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Gra
ph
ing
Po
lyn
om
ial F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-2
7-2
©G
lenc
oe/M
cGra
w-H
ill38
1G
lenc
oe A
lgeb
ra 2
Lesson 7-2
Gra
ph
Po
lyn
om
ial F
un
ctio
ns
Lo
cati
on
Pri
nci
ple
Sup
pose
y�
f(x)
rep
rese
nts
a po
lyno
mia
l fun
ctio
n an
d a
and
bar
e tw
o nu
mbe
rs s
uch
that
f(a)
�0
and
f(b)
�0.
The
n th
e fu
nctio
n ha
s at
leas
t on
e re
al z
ero
betw
een
aan
d b.
Det
erm
ine
the
valu
es o
f x
bet
wee
n w
hic
h e
ach
rea
l ze
ro o
f th
efu
nct
ion
f(x
) �
2x4
�x3
�5
is l
ocat
ed.T
hen
dra
w t
he
grap
h.
Mak
e a
tabl
e of
val
ues
.Loo
k at
th
e va
lues
of
f(x)
to
loca
te t
he
zero
s.T
hen
use
th
e po
ints
to
sket
ch a
gra
ph o
f th
e fu
nct
ion
.T
he
chan
ges
in s
ign
in
dica
te t
hat
th
ere
are
zero
sbe
twee
n x
��
2 an
d x
��
1 an
d be
twee
n x
�1
and
x�
2.
Gra
ph
eac
h f
un
ctio
n b
y m
akin
g a
tab
le o
f va
lues
.Det
erm
ine
the
valu
es o
f x
atw
hic
h o
r b
etw
een
wh
ich
eac
h r
eal
zero
is
loca
ted
.
1.f(
x) �
x3�
2x2
�1
2.f(
x) �
x4�
2x3
�5
3.f(
x) �
�x4
�2x
2�
1
bet
wee
n 0
an
d �
1;b
etw
een
�2
and
�3;
at �
1 at
1;
bet
wee
n 1
an
d 2
bet
wee
n 1
an
d 2
4.f(
x) �
x3�
3x2
�4
5.f(
x) �
3x3
�2x
�1
6.f(
x) �
x4�
3x3
�1
at �
1,2
bet
wee
n 0
an
d 1
bet
wee
n 0
an
d 1
;b
etw
een
2 a
nd
3x
f (x)
Ox
f (x)
Ox
f (x)
O
x
f (x)
Ox
f (x)
O
x
f (x)
O4
8–4
–8
8 4 –4 –8
x
f (x)
O
xf(
x)
�2
35
�1
�2
0�
5
1�
4
219
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill38
2G
lenc
oe A
lgeb
ra 2
Max
imu
m a
nd
Min
imu
m P
oin
tsA
qu
adra
tic
fun
ctio
n h
as e
ith
er a
max
imu
m o
r a
min
imu
m p
oin
t on
its
gra
ph.F
or h
igh
er d
egre
e po
lyn
omia
l fu
nct
ion
s,yo
u c
an f
ind
turn
ing
poin
ts,w
hic
h r
epre
sen
t re
lati
ve m
axim
um
or r
elat
ive
min
imu
mpo
ints
.
Gra
ph
f(x
) �
x3�
6x2
�3.
Est
imat
e th
e x-
coor
din
ates
at
wh
ich
th
ere
lati
ve m
axim
a an
d m
inim
a oc
cur.
Mak
e a
tabl
e of
val
ues
an
d gr
aph
th
e fu
nct
ion
.A
rel
ativ
e m
axim
um
occ
urs
at x
��
4 an
d a
rela
tive
min
imu
m o
ccu
rs a
t x
�0.
Gra
ph
eac
h f
un
ctio
n b
y m
akin
g a
tab
le o
f va
lues
.Est
imat
e th
e x-
coor
din
ates
at
wh
ich
th
e re
lati
ve m
axim
a an
d m
inim
a oc
cur.
1.f(
x) �
x3�
3x2
2.f(
x) �
2x3
�x2
�3x
3.f(
x) �
2x3
�3x
�2
max
.at
0,m
in.a
t 2
max
.ab
ou
t �
1,m
ax.a
bo
ut
�1,
min
.ab
ou
t 0.
5m
in.a
bo
ut
14.
f(x)
�x4
�7x
�3
5.f(
x) �
x5�
2x2
�2
6.f(
x) �
x3�
2x2
�3
min
.ab
ou
t 1
max
.at
0,m
ax.a
bo
ut
�1,
min
.ab
ou
t 1
min
.at
0
x
f (x)
Ox
f (x)
Ox
f (x)
O4
8–4
–8
8 4 –4 –8
x
f (x)
Ox
f (x)
Ox
f (x)
O
x
f (x)
O2
–2–4
24 16 8
← in
dica
tes
a re
lativ
e m
axim
um
← z
ero
betw
een
x�
�1,
x�
0
← in
dica
tes
a re
lativ
e m
inim
um
xf(
x)
�5
22
�4
29
�3
24
�2
13
�1
2
0�
3
14
229
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Gra
ph
ing
Po
lyn
om
ial F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-2
7-2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A6 Glencoe Algebra 2
Answers (Lesson 7-2)
Skil
ls P
ract
ice
Gra
ph
ing
Po
lyn
om
ial F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-2
7-2
©G
lenc
oe/M
cGra
w-H
ill38
3G
lenc
oe A
lgeb
ra 2
Lesson 7-2
Com
ple
te e
ach
of
the
foll
owin
g.a.
Gra
ph
eac
h f
un
ctio
n b
y m
akin
g a
tab
le o
f va
lues
.b
.Det
erm
ine
con
secu
tive
val
ues
of
xb
etw
een
wh
ich
eac
h r
eal
zero
is
loca
ted
.c.
Est
imat
e th
e x-
coor
din
ates
at
wh
ich
th
e re
lati
ve m
axim
a an
d m
inim
a oc
cur.
1.f(
x) �
x3�
3x2
�1
2.f(
x) �
x3�
3x�
1
zero
s b
etw
een
�1
and
0,0
an
d 1
,ze
ros
bet
wee
n �
2 an
d �
1,0
and
1,
and
2 a
nd
3;
rel.
max
.at
x�
0,an
d 1
an
d 2
;re
l.m
ax.a
t x
��
1,re
l.m
in.a
t x
�2
rel.
min
.at
x�
1
3.f(
x) �
2x3
�9x
2�
12x
�2
4.f(
x) �
2x3
�3x
2�
2
zero
bet
wee
n �
1 an
d 0
;ze
ro b
etw
een
�1
and
0;
rel.
max
.at
x�
�2,
rel.
min
.at
x�
1,re
l.m
ax.a
t x
�0
rel.
min
.at
x�
�1
5.f(
x) �
x4�
2x2
�2
6.f(
x) �
0.5x
4�
4x2
�4
zero
s b
etw
een
�2
and
�1,
and
ze
ros
bet
wee
n �
1 an
d �
2,�
2 an
d
1 an
d 2
;re
l.m
ax.a
t x
�0,
�3,
1 an
d 2
,an
d 2
an
d 3
;rel
.max
.at
rel.
min
.at
x�
�1
and
x�
1x
�0,
rel.
min
.at
x�
�2
and
x�
2
x
f (x)
O
xf(
x)
�3
8.5
�2
�4
�1
0.5
04
10.
52
�4
38.
5
x
f (x)
O
xf(
x)
�3
61�
26
�1
�3
0�
21
�3
26
361
x
f (x)
O
xf(
x)
�1
�3
02
11
26
329
x
f (x)
O
xf(
x)
�3
�7
�2
�2
�1
�3
02
125
x
f (x)
O
xf(
x)
�3
�17
�2
�1
�1
30
11
�1
23
319
x
f (x)
O
xf(
x)
�2
�19
�1
�3
01
1�
12
�3
31
417
©G
lenc
oe/M
cGra
w-H
ill38
4G
lenc
oe A
lgeb
ra 2
Com
ple
te e
ach
of
the
foll
owin
g.a.
Gra
ph
eac
h f
un
ctio
n b
y m
akin
g a
tab
le o
f va
lues
.b
.Det
erm
ine
con
secu
tive
val
ues
of
xb
etw
een
wh
ich
eac
h r
eal
zero
is
loca
ted
.c.
Est
imat
e th
e x-
coor
din
ates
at
wh
ich
th
e re
lati
ve a
nd
rel
ativ
e m
inim
a oc
cur.
1.f(
x) �
�x3
�3x
2�
32.
f(x)
�x3
�1.
5x2
�6x
�1
x
f (x)
O8 4 –4 –8
24
–2–4
xf(
x)
�2
�1
�1
4.5
01
1�
5.5
2�
93
�3.
54
17
x
f (x)
O
xf(
x)
�2
17�
11
0�
31
�1
21
3�
34
�19Pra
ctic
e (
Ave
rag
e)
Gra
ph
ing
Po
lyn
om
ial F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-2
7-2
zero
s b
etw
een
�1
zero
s b
etw
een
�2
and
0,1
an
d 2
,an
d �
1,0
and
1,
and
2 a
nd
3;
rel.
max
.at
x�
2,an
d 3
an
d 4
;re
l.m
ax.a
t x
��
1,re
l.m
in.a
t x
�0
rel.
min
.at
x�
2
3.f(
x) �
0.75
x4�
x3�
3x2
�4
4.f(
x) �
x4�
4x3
�6x
2�
4x�
3
zero
s b
etw
een
�3
and
�2,
and
ze
ros
bet
wee
n �
3 an
d �
2,�
2 an
d �
1;re
l.m
ax.a
t x
�0,
and
0 a
nd
1;
rel.
min
.at
x�
�1
rel.
min
.at
x�
�2
and
x�
1
PR
ICE
SF
or E
xerc
ises
5 a
nd
6,u
se t
he
foll
owin
g in
form
atio
n.
Th
e C
onsu
mer
Pri
ce I
nde
x (C
PI)
giv
es t
he
rela
tive
pri
ce
for
a fi
xed
set
of g
oods
an
d se
rvic
es.T
he
CP
I fr
om
Sep
tem
ber,
2000
to
July
,200
1 is
sh
own
in
th
e gr
aph
.So
urce
: U. S
. Bur
eau
of L
abor
Sta
tistic
s
5.D
escr
ibe
the
turn
ing
poin
ts o
f th
e gr
aph
.re
l max
.in
Nov
.an
d J
un
e;re
l.m
in in
Dec
.6.
If t
he
grap
h w
ere
mod
eled
by
a po
lyn
omia
l eq
uat
ion
,w
hat
is
the
leas
t de
gree
th
e eq
uat
ion
cou
ld h
ave?
4
7.LA
BO
RA
tow
n’s
jobl
ess
rate
can
be
mod
eled
by
(1,3
.3),
(2,4
.9),
(3,5
.3),
(4,6
.4),
(5,4
.5),
(6,5
.6),
(7,2
.5),
(8,2
.7).
How
man
y tu
rnin
g po
ints
wou
ld t
he
grap
h o
f a
poly
nom
ial
fun
ctio
n t
hro
ugh
th
ese
poin
ts h
ave?
Des
crib
e th
em.
4:2
rel.
max
.an
d 2
rel
.min
.
Mo
nth
s Si
nce
Sep
tem
ber
, 200
0
Consumer Price Index
20
46
13
57
89
1011
179
178
177
176
175
174
173
f(x)
xO
xf(
x)
�3
12�
2�
3�
1�
40
�3
112
277
f(x)
xO
xf(
x)
�3
10.7
5�
2�
4�
10.
750
41
2.75
212
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
An
swer
s
Answers (Lesson 7-2)
Readin
g t
o L
earn
Math
em
ati
csG
rap
hin
g P
oly
no
mia
l Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-2
7-2
©G
lenc
oe/M
cGra
w-H
ill38
5G
lenc
oe A
lgeb
ra 2
Lesson 7-2
Pre-
Act
ivit
yH
ow c
an g
rap
hs
of p
olyn
omia
l fu
nct
ion
s sh
ow t
ren
ds
in d
ata?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-2
at
the
top
of p
age
353
in y
our
text
book
.
Th
ree
poin
ts o
n t
he
grap
h s
how
n i
n y
our
text
book
are
(0,
14),
(70,
3.78
),an
d(1
00,9
).G
ive
the
real
-wor
ld m
ean
ing
of t
he
coor
din
ates
of
thes
e po
ints
.S
amp
le a
nsw
er:
In 1
900,
14%
of
the
U.S
.po
pu
lati
on
was
fore
ign
bo
rn.I
n 1
970,
3.78
% o
f th
e p
op
ula
tio
n w
as f
ore
ign
bo
rn.I
n 2
000,
9% o
f th
e p
op
ula
tio
n w
as f
ore
ign
bo
rn.
Rea
din
g t
he
Less
on
1.S
upp
ose
that
f(x
) is
a t
hir
d-de
gree
pol
ynom
ial
fun
ctio
n a
nd
that
can
d d
are
real
nu
mbe
rs,w
ith
d�
c.In
dica
te w
het
her
eac
h s
tate
men
t is
tru
eor
fal
se.(
Rem
embe
r th
attr
ue
mea
ns
alw
ays
tru
e.)
a.If
f(c
) �
0 an
d f(
d)
�0,
ther
e is
exa
ctly
on
e re
al z
ero
betw
een
can
d d
.fa
lse
b.
If f
(c)
�f(
d)
0,
ther
e ar
e n
o re
al z
eros
bet
wee
n c
and
d.
fals
e
c.If
f(c
) �
0 an
d f(
d)
�0,
ther
e is
at
leas
t on
e re
al z
ero
betw
een
can
d d
.tr
ue
2.M
atch
eac
h g
raph
wit
h i
ts d
escr
ipti
on.
a.th
ird-
degr
ee p
olyn
omia
l w
ith
on
e re
lati
ve m
axim
um
an
d on
e re
lati
ve m
inim
um
;le
adin
g co
effi
cien
t n
egat
ive
iii
b.
fou
rth
-deg
ree
poly
nom
ial
wit
h t
wo
rela
tive
min
ima
and
one
rela
tive
max
imu
mi
c.th
ird-
degr
ee p
olyn
omia
l w
ith
on
e re
lati
ve m
axim
um
an
d on
e re
lati
ve m
inim
um
;le
adin
g co
effi
cien
t po
siti
veiv
d.
fou
rth
-deg
ree
poly
nom
ial
wit
h t
wo
rela
tive
max
ima
and
one
rela
tive
min
imu
mii
i.ii
.ii
i.iv
.
Hel
pin
g Y
ou
Rem
emb
er
3.T
he
orig
ins
of w
ords
can
hel
p yo
u t
o re
mem
ber
thei
r m
ean
ing
and
to d
isti
ngu
ish
betw
een
sim
ilar
wor
ds.L
ook
up
max
imu
man
d m
inim
um
in a
dic
tion
ary
and
desc
ribe
thei
r or
igin
s (o
rigi
nal
lan
guag
e an
d m
ean
ing)
.S
amp
le a
nsw
er:
Max
imu
mco
mes
fro
m t
he
Lat
in w
ord
max
imu
s,m
ean
ing
gre
ates
t.M
inim
um
com
es f
rom
the
Lat
in w
ord
min
imu
s,m
ean
ing
leas
t.
x
f (x)
Ox
f (x)
Ox
f (x)
Ox
f (x)
O
©G
lenc
oe/M
cGra
w-H
ill38
6G
lenc
oe A
lgeb
ra 2
Go
lden
Rec
tan
gle
sU
se a
str
aigh
ted
ge,a
com
pas
s,an
d t
he
inst
ruct
ion
s b
elow
to
con
stru
ct
a go
lden
rec
tan
gle.
1.C
onst
ruct
squ
are
AB
CD
wit
h s
ides
of
2 ce
nti
met
ers.
2.C
onst
ruct
th
e m
idpo
int
of A �
B�.C
all
the
mid
poin
t M
.
3.U
sin
g M
as t
he
cen
ter,
set
you
r co
mpa
ss
open
ing
at M
C.C
onst
ruct
an
arc
wit
h
cen
ter
Mth
at i
nte
rsec
ts A �
B�.C
all
the
poin
t of
in
ters
ecti
on P
.
4.C
onst
ruct
a l
ine
thro
ugh
Pth
at i
s pe
rpen
dicu
lar
to A �
B�.
5.E
xten
d D�
C�so
th
at i
t in
ters
ects
th
e pe
rpen
dicu
lar.
Cal
l th
e in
ters
ecti
on p
oin
t Q
.A
PQ
Dis
a g
olde
n r
ecta
ngl
e.C
hec
k th
is
con
clu
sion
by
fin
din
g th
e va
lue
of �Q A
PP �.
0.62
A f
igu
re c
onsi
stin
g of
sim
ilar
gol
den
rec
tan
gles
is
show
n b
elow
.Use
a
com
pas
s an
d t
he
inst
ruct
ion
s b
elow
to
dra
w q
uar
ter-
circ
le a
rcs
that
fo
rm a
sp
iral
lik
e th
at f
oun
d i
n t
he
shel
l of
a c
ham
ber
ed n
auti
lus.
6.U
sin
g A
as a
cen
ter,
draw
an
arc
th
at p
asse
s th
rou
gh
Ban
d C
.
7.U
sin
g D
as a
cen
ter,
draw
an
arc
th
at p
asse
s th
rou
gh
Can
d E
.
8.U
sin
g F
as a
cen
ter,
draw
an
arc
th
at p
asse
s th
rou
gh
Ean
d G
.
9.C
onti
nu
e dr
awin
g ar
cs,
usi
ng
H,K
,an
d M
as
the
cen
ters
.
C
BA
G
HJD
E
K
M
LF
D AM
QC
PB
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-2
7-2
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 7-3)
Stu
dy G
uid
e a
nd I
nte
rven
tion
So
lvin
g E
qu
atio
ns
Usi
ng
Qu
adra
tic
Tech
niq
ues
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-3
7-3
©G
lenc
oe/M
cGra
w-H
ill38
7G
lenc
oe A
lgeb
ra 2
Lesson 7-3
Qu
adra
tic
Form
Cer
tain
pol
ynom
ial
expr
essi
ons
in x
can
be
wri
tten
in
th
e qu
adra
tic
form
au
2�
bu�
cfo
r an
y n
um
bers
a,b
,an
d c,
a
0,w
her
e u
is a
n e
xpre
ssio
n i
n x
.
Wri
te e
ach
pol
ynom
ial
in q
uad
rati
c fo
rm,i
f p
ossi
ble
.
a.3a
6�
9a3
�12
Let
u�
a3.
3a6
�9a
3�
12 �
3(a3
)2�
9(a3
) �
12
b.
101b
�49
�b�
�42
Let
u�
�b�.
101b
�49
�b�
�42
�10
1(�
b�)2
�49
(�b�)
�42
c.24
a5
�12
a3
�18
Th
is e
xpre
ssio
n c
ann
ot b
e w
ritt
en i
n q
uad
rati
c fo
rm,s
ince
a5
(a
3 )2 .
Wri
te e
ach
pol
ynom
ial
in q
uad
rati
c fo
rm,i
f p
ossi
ble
.
1.x4
�6x
2�
82.
4p4
�6p
2�
8
(x2 )
2�
6(x
2 ) �
84(
p2 )
2�
6(p
2 ) �
8
3.x8
�2x
4�
14.
x�1 8��
2x� 11 6�
�1
(x4 )
2�
2(x
4 ) �
1� x� 11 6�
�2�
2� x� 11 6�
� �1
5.6x
4�
3x3
�18
6.12
x4�
10x2
�4
no
t p
oss
ible
12(x
2 )2
�10
(x2 )
�4
7.24
x8�
x4�
48.
18x6
�2x
3�
12
24(x
4 )2
�x
4�
418
(x3 )
2�
2(x
3 ) �
12
9.10
0x4
�9x
2�
1510
.25x
8�
36x6
�49
100(
x2 )
2�
9(x
2 ) �
15n
ot
po
ssib
le
11.4
8x6
�32
x3�
2012
.63x
8�
5x4
�29
48(x
3 )2
�32
(x3 )
�20
63(x
4 )2
�5(
x4 )
�29
13.3
2x10
�14
x5�
143
14.5
0x3
�15
x�x�
�18
32(x
5 )2
�14
(x5 )
�14
350
� x�3 2� �2�
15� x�3 2� � �
18
15.6
0x6
�7x
3�
316
.10x
10�
7x5
�7
60(x
3 )2
�7(
x3 )
�3
10(x
5 )2
�7(
x5 )
�7
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill38
8G
lenc
oe A
lgeb
ra 2
Solv
e Eq
uat
ion
s U
sin
g Q
uad
rati
c Fo
rmIf
a p
olyn
omia
l ex
pres
sion
can
be
wri
tten
in q
uad
rati
c fo
rm,t
hen
you
can
use
wh
at y
ou k
now
abo
ut
solv
ing
quad
rati
c eq
uat
ion
s to
solv
e th
e re
late
d po
lyn
omia
l eq
uat
ion
.
Sol
ve x
4�
40x
2�
144
�0.
x4�
40x2
�14
4 �
0O
rigin
al e
quat
ion
(x2 )
2�
40(x
2 ) �
144
�0
Writ
e th
e ex
pres
sion
on
the
left
in q
uadr
atic
for
m.
(x2
�4)
(x2
�36
) �
0F
acto
r.x2
�4
�0
orx2
�36
�0
Zer
o P
rodu
ct P
rope
rty
(x�
2)(x
�2)
�0
or(x
�6)
(x�
6) �
0F
acto
r.
x�
2 �
0or
x�
2 �
0or
x�
6 �
0or
x�
6 �
0Z
ero
Pro
duct
Pro
pert
y
x�
2or
x�
�2
orx
�6
orx
��
6S
impl
ify.
Th
e so
luti
ons
are
2
and
6.
Sol
ve 2
x�
�x�
�15
�0.
2x�
�x�
�15
�0
Orig
inal
equ
atio
n
2(�
x�)2
��
x��
15 �
0W
rite
the
expr
essi
on o
n th
e le
ft in
qua
drat
ic f
orm
.
(2�
x��
5)(�
x��
3) �
0F
acto
r.
2�x�
�5
�0
or�
x��
3 �
0Z
ero
Pro
duct
Pro
pert
y
�x�
�or
�x�
��
3S
impl
ify.
Sin
ce t
he
prin
cipa
l sq
uar
e ro
ot o
f a
nu
mbe
r ca
nn
ot b
e n
egat
ive,
�x�
��
3 h
as n
o so
luti
on.
Th
e so
luti
on i
s or
6.
Sol
ve e
ach
eq
uat
ion
.
1.x4
�49
2.x4
�6x
2�
�8
3.x4
�3x
2�
54
��
7�,�
i�7�
�2,
��
2��
3,�
i�6�
4.3t
6�
48t2
�0
5.m
6�
16m
3�
64 �
06.
y4�
5y2
�4
�0
0,�
2,�
2i2,
�1
�i�
3��
1,�
2
7.x4
�29
x2�
100
�0
8.4x
4�
73x2
�14
4 �
09.
��
12 �
0
�5,
�2
�4,
�,
10.x
�5�
x��
6 �
011
.x�
10�
x��
21 �
012
.x�2 3�
�5x
�1 3��
6 �
0
4,9
9,49
27,81 � 4
1 � 33 � 2
7 � x1 � x2
1 � 425 � 4
5 � 2Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
So
lvin
g E
qu
atio
ns
Usi
ng
Qu
adra
tic
Tech
niq
ues
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-3
7-3
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A9 Glencoe Algebra 2
An
swer
s
Answers (Lesson 7-3)
Skil
ls P
ract
ice
So
lvin
g E
qu
atio
ns
Usi
ng
Qu
adra
tic
Tech
niq
ues
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-3
7-3
©G
lenc
oe/M
cGra
w-H
ill38
9G
lenc
oe A
lgeb
ra 2
Lesson 7-3
Wri
te e
ach
exp
ress
ion
in
qu
adra
tic
form
,if
pos
sib
le.
1.5x
4�
2x2
�8
5(x
2 )2
�2(
x2 )
�8
2.3y
8�
4y2
�3
no
t p
oss
ible
3.10
0a6
�a3
100(
a3)2
�a3
4.x8
�4x
4�
9(x
4 )2
�4(
x4 )
�9
5.12
x4�
7x2
12(x
2 )2
�7(
x2 )
6.6b
5�
3b3
�1
no
t p
oss
ible
7.15
v6�
8v3
�9
15(v
3 )2
�8(
v3 )
�9
8.a9
�5a
5�
7aa[
(a4 )
2�
5(a4
) �
7]
Sol
ve e
ach
eq
uat
ion
.
9.a3
�9a
2�
14a
�0
0,7,
210
.x3
�3x
20,
3
11.t
4�
3t3
�40
t2�
00,
�5,
812
.b3
�8b
2�
16b
�0
0,4
13.m
4�
4�
�2�,
�2�,
�i�
2�,i�
2�14
.w3
�6w
�0
0,�
6�,�
�6�
15.m
4�
18m
2�
�81
�3,
316
.x5
�81
x�
00,
�3,
3,�
3i,3
i
17.h
4�
10h
2�
�9
�1,
1,�
3,3
18.a
4�
9a2
�20
�0
�2,
2,�
5�,�
�5�
19.y
4�
7y2
�12
�0
20.v
4�
12v2
�35
�0
2,�
2,�
3�,�
�3�
�5�,
��
5�,�
7�,�
�7�
21.x
5�
7x3
�6x
�0
22.c
�2 3��
7c�1 3�
�12
�0
0,�
1,1,
�6�,
��
6��
64,�
27
23.z
�5�
z��
�6
4,9
24.x
�30
�x�
�20
0 �
010
0,40
0
©G
lenc
oe/M
cGra
w-H
ill39
0G
lenc
oe A
lgeb
ra 2
Wri
te e
ach
exp
ress
ion
in
qu
adra
tic
form
,if
pos
sib
le.
1.10
b4�
3b2
�11
2.�
5x8
�x2
�6
3.28
d6
�25
d3
10(b
2 )2
�3(
b2 )
�11
no
t p
oss
ible
28(d
3 )2
�25
(d3 )
4.4s
8�
4s4
�7
5.50
0x4
�x2
6.8b
5�
8b3
�1
4(s4
)2�
4(s4
) �
750
0(x2
)2�
x2n
ot
po
ssib
le
7.32
w5
�56
w3
�8w
8.e�2 3�
�7e
�1 3��
109.
x�1 5��
29x� 11 0�
� 2
8w[4
(w2 )
2�
7(w
2 ) �
1]( e�1 3� )2
�7( e
�1 3� ) �10
( x� 11 0�)2
�29
( x� 11 0�) �
2
Sol
ve e
ach
eq
uat
ion
.
10.y
4�
7y3
�18
y2�
0�
2,0,
911
.s5
�4s
4�
32s3
�0
�8,
0,4
12.m
4�
625
�0
�5,
5,�
5i,5
i13
.n4
�49
n2
�0
0,�
7,7
14.x
4�
50x2
�49
�0
�1,
1,�
7,7
15.t
4�
21t2
�80
�0
�4,
4,�
5�,�
�5�
16.4
r6�
9r4
�0
0,�3 2� ,
��3 2�
17.x
4�
24 �
�2x
2�
2,2,
�i�
6�,i�
6�
18.d
4�
16d
2�
48 �
2,2,
�2�
3�,2�
3�19
.t3
�34
3 �
07,
,
20.x
�1 2��
5x�1 4�
�6
�0
16,8
121
.x�4 3�
�29
x�2 3��
100
�0
8,12
5
22.y
3�
28y�3 2�
�27
�0
1,9
23.n
�10
�n�
�25
�0
25
24.w
�12
�w�
�27
�0
9,81
25.x
�2�
x��
80 �
010
0
26.P
HY
SIC
SA
pro
ton
in
a m
agn
etic
fie
ld f
ollo
ws
a pa
th o
n a
coo
rdin
ate
grid
mod
eled
by
the
fun
ctio
n f
(x)
�x4
�2x
2�
15.W
hat
are
th
e x-
coor
din
ates
of
the
poin
ts o
n t
he
grid
wh
ere
the
prot
on c
ross
es t
he
x-ax
is?
��
5� ,�
5�
27.S
UR
VEY
ING
Vis
ta c
oun
ty i
s se
ttin
g as
ide
a la
rge
parc
el o
f la
nd
to p
rese
rve
it a
s op
ensp
ace.
Th
e co
un
ty h
as h
ired
Meg
han
’s s
urv
eyin
g fi
rm t
o su
rvey
th
e pa
rcel
,wh
ich
is
inth
e sh
ape
of a
rig
ht
tria
ngl
e.T
he
lon
ger
leg
of t
he
tria
ngl
e m
easu
res
5 m
iles
les
s th
anth
e sq
uar
e of
th
e sh
orte
r le
g,an
d th
e h
ypot
enu
se o
f th
e tr
ian
gle
mea
sure
s 13
mil
es l
ess
than
tw
ice
the
squ
are
of t
he
shor
ter
leg.
Th
e le
ngt
h o
f ea
ch b
oun
dary
is
a w
hol
e n
um
ber.
Fin
d th
e le
ngt
h o
f ea
ch b
oun
dary
.3
mi,
4 m
i,5
mi
�7
�7i
�3�
�� 2
�7
�7i
�3�
�� 2
Pra
ctic
e (
Ave
rag
e)
So
lvin
g E
qu
atio
ns
Usi
ng
Qu
adra
tic
Tech
niq
ues
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-3
7-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 7-3)
Readin
g t
o L
earn
Math
em
ati
csS
olv
ing
Eq
uat
ion
s U
sin
g Q
uad
rati
c Te
chn
iqu
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-3
7-3
©G
lenc
oe/M
cGra
w-H
ill39
1G
lenc
oe A
lgeb
ra 2
Lesson 7-3
Pre-
Act
ivit
yH
ow c
an s
olvi
ng
pol
ynom
ial
equ
atio
ns
hel
p y
ou t
o fi
nd
dim
ensi
ons?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-3
at
the
top
of p
age
360
in y
our
text
book
.
Exp
lain
how
th
e fo
rmu
la g
iven
for
th
e vo
lum
e of
th
e bo
x ca
n b
e ob
tain
edfr
om t
he
dim
ensi
ons
show
n i
n t
he
figu
re.
Sam
ple
an
swer
:Th
e vo
lum
e o
f a
rect
ang
ula
r b
ox is
giv
en
by t
he
form
ula
V�
�wh
.Su
bst
itu
te 5
0 �
2xfo
r �,
32 �
2xfo
r w
,an
d x
for
hto
get
V
(x)
�(5
0 �
2x)(
32 �
2x)(
x) �
4x3
�16
4x2
�16
00x.
Rea
din
g t
he
Less
on
1.W
hic
h o
f th
e fo
llow
ing
expr
essi
ons
can
be
wri
tten
in
qu
adra
tic
form
?b
,c,d
,f,g
,h,i
a.x3
�6x
2�
9b
.x4
�7x
2�
6c.
m6
�4m
3�
4
d.
y�
2y�1 2�
�15
e.x5
�x3
�1
f.r4
�6
�r8
g.p�1 4�
�8p
�1 2��
12h
.r�1 3�
�2r
�1 6��
3i.
5�z�
�2z
�3
2.M
atch
eac
h e
xpre
ssio
n f
rom
th
e li
st o
n t
he
left
wit
h i
ts f
acto
riza
tion
fro
m t
he
list
on
th
e ri
ght.
a.x4
�3x
2�
40vi
i.(x
3�
3)(x
3�
3)
b.
x4�
10x2
�25
vii
.(�
x��
3)(�
x��
3)
c.x6
�9
iii
i.(�
x��
3)2
d.
x�
9ii
iv.
(x2
�1)
(x4
�x2
�1)
e.x6
�1
ivv.
(x2
�5)
2
f.x
�6�
x��
9iii
vi.
(x2
�5)
(x2
�8)
Hel
pin
g Y
ou
Rem
emb
er
3.W
hat
is
an e
asy
way
to
tell
wh
eth
er a
tri
nom
ial
in o
ne
vari
able
con
tain
ing
one
con
stan
tte
rm c
an b
e w
ritt
en i
n q
uad
rati
c fo
rm?
Sam
ple
an
swer
:L
oo
k at
th
e tw
o t
erm
s th
at a
re n
ot
con
stan
ts a
nd
com
par
e th
e ex
po
nen
ts o
n t
he
vari
able
.If
on
e o
f th
e ex
po
nen
ts is
tw
ice
the
oth
er,t
he
trin
om
ial c
an b
e w
ritt
en in
qu
adra
tic
form
.
©G
lenc
oe/M
cGra
w-H
ill39
2G
lenc
oe A
lgeb
ra 2
Od
d a
nd
Eve
n P
oly
no
mia
l Fu
nct
ion
s
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-3
7-3
Fu
nct
ion
s w
hos
e gr
aph
s ar
e sy
mm
etri
c w
ith
resp
ect
to t
he o
rigi
n ar
e ca
lled
odd
func
tion
s.If
f(�
x) �
�f(
x) f
or a
ll x
in t
he d
omai
n of
f(x
),th
en f
(x)
is o
dd.
Fu
nct
ion
s w
hos
e gr
aph
s ar
e sy
mm
etri
c w
ith
resp
ect
to t
he y
-axi
s ar
e ca
lled
even
func
tion
s.If
f(�
x) �
f(x)
for
all
xin
th
e do
mai
n o
f f(
x),
then
f(x
) is
eve
n.
x
f (x)
O1
2–2
–1
6 4 2f(
x) �
1 – 4x4 �
4
x
f (x)
O1
2–2
–1
4 2 –2 –4
f(x)
� 1 – 2x3
Exam
ple
Exam
ple
Det
erm
ine
wh
eth
er f
(x)
�x3
�3x
is o
dd
,eve
n,o
r n
eith
er.
f(x)
�x3
�3x
f(�
x) �
(�x)
3�
3(�
x)R
epla
ce x
with
�x.
��
x3�
3xS
impl
ify.
��
(x3
�3x
)F
acto
r ou
t �
1.
��
f(x)
Sub
stut
ute.
Th
eref
ore,
f(x)
is
odd.
Th
e gr
aph
at
the
righ
t ve
rifi
es t
hat
f(x
) is
odd
.T
he
grap
h o
f th
e fu
nct
ion
is
sym
met
ric
wit
h
resp
ect
to t
he
orig
in.
Det
erm
ine
wh
eth
er e
ach
fu
nct
ion
is
odd
,eve
n,o
r n
eith
erb
y gr
aph
ing
or b
y ap
ply
ing
the
rule
s fo
r od
d a
nd
eve
n f
un
ctio
ns.
1.f(
x) �
4x2
even
2.f(
x) �
�7x
4ev
en
3.f(
x) �
x7o
dd
4.f(
x) �
x3�
x2n
eith
er
5.f(
x) �
3x3
�1
nei
ther
6.f(
x) �
x8�
x5�
6n
eith
er
7.f(
x) �
�8x
5�
2x3
�6x
od
d8.
f(x)
�x4
�3x
3�
2x2
�6x
�1
nei
ther
9.f(
x) �
x4�
3x2
�11
even
10.f
(x)
�x7
�6x
5�
2x3
�x
od
d
11.C
ompl
ete
the
foll
owin
g de
fin
itio
ns:
A p
olyn
omia
l fu
nct
ion
is
odd
if a
nd
only
if a
ll t
he
term
s ar
e of
de
gree
s.A
pol
ynom
ial
fun
ctio
n i
s ev
en
if a
nd
only
if
all
the
term
s ar
e of
de
gree
s.ev
eno
dd
x
f (x)
O1
2–2
–1
4 2 –2 –4
f(x)
� x
3 �
3x
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
An
swer
s
Answers (Lesson 7-4)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Th
e R
emai
nd
er a
nd
Fac
tor T
heo
rem
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-4
7-4
©G
lenc
oe/M
cGra
w-H
ill39
3G
lenc
oe A
lgeb
ra 2
Lesson 7-4
Syn
thet
ic S
ub
stit
uti
on
Rem
ain
der
The
rem
aind
er,
whe
n yo
u di
vide
the
pol
ynom
ial f
(x)
by (
x�
a),
is t
he c
onst
ant
f(a)
.T
heo
rem
f(x)
�q
(x)
�(x
�a)
�f(
a),
whe
re q
(x)
is a
pol
ynom
ial w
ith d
egre
e on
e le
ss t
han
the
degr
ee o
f f(
x).
If f
(x)
�3x
4�
2x3
�5x
2�
x�
2,fi
nd
f(�
2).
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Met
hod
1S
ynth
etic
Su
bsti
tuti
onB
y th
e R
emai
nde
r T
heo
rem
,f(�
2) s
hou
ldbe
th
e re
mai
nde
r w
hen
you
div
ide
the
poly
nom
ial
by x
�2.
�2
32
�5
1�
2�
68
�6
103
�4
3�
58
Th
e re
mai
nde
r is
8,s
o f(
�2)
�8.
Met
hod
2D
irec
t S
ubs
titu
tion
Rep
lace
xw
ith
�2.
f(x)
�3x
4�
2x3
�5x
2�
x�
2f(
�2)
�3(
�2)
4�
2(�
2)3
�5(
�2)
2�
(�2)
�2
�48
�16
�20
�2
�2
or 8
So
f(�
2) �
8.
If f
(x)
�5x
3�
2x�
1,fi
nd
f(3
).A
gain
,by
the
Rem
ain
der
Th
eore
m,f
(3)
shou
ld b
e th
e re
mai
nde
r w
hen
you
div
ide
the
poly
nom
ial
by x
�3.
35
02
�1
1545
141
515
4714
0T
he
rem
ain
der
is 1
40,s
o f(
3) �
140.
Use
syn
thet
ic s
ub
stit
uti
on t
o fi
nd
f(�
5) a
nd
f�
�for
each
fu
nct
ion
.
1.f(
x) �
�3x
2�
5x�
1�
101;
2.f(
x) �
4x2
�6x
�7
63;
�3
3.f(
x) �
�x3
�3x
2�
519
5;�
4.f(
x) �
x4�
11x2
�1
899;
Use
syn
thet
ic s
ub
stit
uti
on t
o fi
nd
f(4
) an
d f
(�3)
for
eac
h f
un
ctio
n.
5.f(
x) �
2x3
�x2
�5x
�3
6.f(
x) �
3x3
�4x
�2
127;
�27
178;
�67
7.f(
x) �
5x3
�4x
2�
28.
f(x)
�2x
4�
4x3
�3x
2�
x�
625
8;�
169
302;
288
9.f(
x) �
5x4
�3x
3�
4x2
�2x
�4
10.f
(x)
�3x
4�
2x3
�x2
�2x
�5
1404
;29
862
7;27
7
11.f
(x)
�2x
4�
4x3
�x2
�6x
�3
12.f
(x)
�4x
4�
4x3
�3x
2�
2x�
321
9;28
280
5;46
2
29 � 1635 � 83 � 4
1 � 2
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill39
4G
lenc
oe A
lgeb
ra 2
Fact
ors
of
Poly
no
mia
lsT
he
Fac
tor
Th
eore
mca
n h
elp
you
fin
d al
l th
e fa
ctor
s of
apo
lyn
omia
l.
Fact
or T
heo
rem
The
bin
omia
l x�
ais
a f
acto
r of
the
pol
ynom
ial f
(x)
if an
d on
ly if
f(a
) �
0.
Sh
ow t
hat
x�
5 is
a f
acto
r of
x3
�2x
2�
13x
�10
.Th
en f
ind
th
ere
mai
nin
g fa
ctor
s of
th
e p
olyn
omia
l.B
y th
e Fa
ctor
Th
eore
m,t
he
bin
omia
l x
�5
is a
fac
tor
of t
he
poly
nom
ial
if �
5 is
a z
ero
of t
he
poly
nom
ial
fun
ctio
n.T
o ch
eck
this
,use
syn
thet
ic s
ubs
titu
tion
.
�5
12
�13
10�
515
�10
1�
32
0
Sin
ce t
he
rem
ain
der
is 0
,x�
5 is
a f
acto
r of
th
e po
lyn
omia
l.T
he
poly
nom
ial
x3�
2x2
�13
x�
10 c
an b
e fa
ctor
ed a
s (x
�5)
(x2
�3x
�2)
.Th
e de
pres
sed
poly
nom
ial
x2�
3x�
2 ca
n b
e fa
ctor
ed a
s (x
�2)
(x�
1).
So
x3�
2x2
�13
x�
10 �
(x�
5)(x
�2)
(x�
1).
Giv
en a
pol
ynom
ial
and
on
e of
its
fac
tors
,fin
d t
he
rem
ain
ing
fact
ors
of t
he
pol
ynom
ial.
Som
e fa
ctor
s m
ay n
ot b
e b
inom
ials
.
1.x3
�x2
�10
x�
8;x
�2
2.x3
�4x
2�
11x
�30
;x�
3(x
�4)
(x�
1)(x
�5)
(x�
2)
3.x3
�15
x2�
71x
�10
5;x
�7
4.x3
�7x
2�
26x
�72
;x�
4(x
�3)
(x�
5)(x
�2)
(x�
9)
5.2x
3�
x2�
7x�
6;x
�1
6.3x
3�
x2�
62x
�40
;x�
4(2
x�
3)(x
�2)
(3x
�2)
(x�
5)
7.12
x3�
71x2
�57
x�
10;x
�5
8.14
x3�
x2�
24x
�9;
x�
1(4
x�
1)(3
x�
2)(7
x�
3)(2
x�
3)
9.x3
�x
�10
;x�
210
.2x3
�11
x2�
19x
�28
;x�
4(x
2�
2x�
5)(2
x2
�3x
�7)
11.3
x3�
13x2
�34
x�
24;x
�6
12.x
4�
x3�
11x2
�9x
�18
;x�
1(3
x2
�5x
�4)
(x�
2)(x
�3)
(x�
3)
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Th
e R
emai
nd
er a
nd
Fac
tor T
heo
rem
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-4
7-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A12 Glencoe Algebra 2
Answers (Lesson 7-4)
Skil
ls P
ract
ice
Th
e R
emai
nd
er a
nd
Fac
tor T
heo
rem
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-4
7-4
©G
lenc
oe/M
cGra
w-H
ill39
5G
lenc
oe A
lgeb
ra 2
Lesson 7-4
Use
syn
thet
ic s
ub
stit
uti
on t
o fi
nd
f(2
) an
d f
(�1)
for
eac
h f
un
ctio
n.
1.f(
x) �
x2�
6x�
521
,02.
f(x)
�x2
�x
�1
3,3
3.f(
x) �
x2�
2x�
2�
2,1
4.f(
x) �
x3�
2x2
�5
21,6
5.f(
x) �
x3�
x2�
2x�
33,
36.
f(x)
�x3
�6x
2�
x�
430
,0
7.f(
x) �
x3�
3x2
�x
�2
�4,
�7
8.f(
x) �
x3�
5x2
�x
�6
�8,
1
9.f(
x) �
x4�
2x2
�9
15,�
610
.f(x
) �
x4�
3x3
�2x
2�
2x�
62,
14
11.f
(x)
�x5
�7x
3�
4x�
1012
.f(x
) �
x6�
2x5
�x4
�x3
�9x
2�
20�
22,2
0�
32,�
26
Giv
en a
pol
ynom
ial
and
on
e of
its
fac
tors
,fin
d t
he
rem
ain
ing
fact
ors
of t
he
pol
ynom
ial.
Som
e fa
ctor
s m
ay n
ot b
e b
inom
ials
.
13.x
3�
2x2
�x
�2;
x�
114
.x3
�x2
�5x
�3;
x�
1
x�
1,x
�2
x�
1,x
�3
15.x
3�
3x2
�4x
�12
;x�
316
.x3
�6x
2�
11x
�6;
x�
3
x�
2,x
�2
x�
1,x
�2
17.x
3�
2x2
�33
x�
90;x
�5
18.x
3�
6x2
�32
;x�
4
x�
3,x
�6
x�
4,x
�2
19.x
3�
x2�
10x
�8;
x�
220
.x3
�19
x�
30;x
�2
x�
1,x
�4
x�
5,x
�3
21.2
x3�
x2�
2x�
1;x
�1
22.2
x3�
x2�
5x�
2;x
�2
2x�
1,x
�1
x�
1,2x
�1
23.3
x3�
4x2
�5x
�2;
3x�
124
.3x3
�x2
�x
�2;
3x�
2
x�
1,x
�2
x2
�x
�1
©G
lenc
oe/M
cGra
w-H
ill39
6G
lenc
oe A
lgeb
ra 2
Use
syn
thet
ic s
ub
stit
uti
on t
o fi
nd
f(�
3) a
nd
f(4
) fo
r ea
ch f
un
ctio
n.
1.f(
x) �
x2�
2x�
36,
272.
f(x)
�x2
�5x
�10
34,6
3.f(
x) �
x2�
5x�
420
,�8
4.f(
x) �
x3�
x2�
2x�
3�
27,4
3
5.f(
x) �
x3�
2x2
�5
�4,
101
6.f(
x) �
x3�
6x2
�2x
�87
,�24
7.f(
x) �
x3�
2x2
�2x
�8
�31
,32
8.f(
x) �
x3�
x2�
4x�
4�
52,6
0
9.f(
x) �
x3�
3x2
�2x
�50
�56
,70
10.f
(x)
�x4
�x3
�3x
2�
x�
1242
,280
11.f
(x)
�x4
�2x
2�
x�
773
,227
12.f
(x)
�2x
4�
3x3
�4x
2�
2x�
128
6,37
7
13.f
(x)
�2x
4�
x3�
2x2
�26
181,
454
14.f
(x)
�3x
4�
4x3
�3x
2�
5x�
339
0,53
7
15.f
(x)
�x5
�7x
3�
4x�
1016
.f(x
) �
x6�
2x5
�x4
�x3
�9x
2�
20�
430,
1446
74,5
828
Giv
en a
pol
ynom
ial
and
on
e of
its
fac
tors
,fin
d t
he
rem
ain
ing
fact
ors
of t
he
pol
ynom
ial.
Som
e fa
ctor
s m
ay n
ot b
e b
inom
ials
.
17.x
3�
3x2
�6x
�8;
x�
218
.x3
�7x
2�
7x�
15;x
�1
x�
1,x
�4
x�
3,x
�5
19.x
3�
9x2
�27
x�
27;x
�3
20.x
3�
x2�
8x�
12;x
�3
x�
3,x
�3
x�
2,x
�2
21.x
3�
5x2
�2x
�24
;x�
222
.x3
�x2
�14
x�
24;x
�4
x�
3,x
�4
x�
3,x
�2
23.3
x3�
4x2
�17
x�
6;x
�2
24.4
x3�
12x2
�x
�3;
x�
3
x�
3,3x
�1
2x�
1,2x
�1
25.1
8x3
�9x
2�
2x�
1;2x
�1
26.6
x3�
5x2
�3x
�2;
3x�
2
3x�
1,3x
�1
2x�
1,x
�1
27.x
5�
x4�
5x3
�5x
2�
4x�
4;x
�1
28.x
5�
2x4
�4x
3�
8x2
�5x
�10
;x�
2
x�
1,x
�1,
x�
2,x
�2
x�
1,x
�1,
x2�
5
29.P
OPU
LATI
ON
Th
e pr
ojec
ted
popu
lati
on i
n t
hou
san
ds f
or a
cit
y ov
er t
he
nex
t se
vera
lye
ars
can
be
esti
mat
ed b
y th
e fu
nct
ion
P(x
) �
x3�
2x2
�8x
�52
0,w
her
e x
is t
he
nu
mbe
r of
yea
rs s
ince
200
0.U
se s
ynth
etic
su
bsti
tuti
on t
o es
tim
ate
the
popu
lati
on
for
2005
.65
5,00
0
30.V
OLU
ME
Th
e vo
lum
e of
wat
er i
n a
rec
tan
gula
r sw
imm
ing
pool
can
be
mod
eled
by
the
poly
nom
ial
2x3
�9x
2�
7x�
6.If
th
e de
pth
of
the
pool
is
give
n b
y th
e po
lyn
omia
l 2x
�1,
wh
at p
olyn
omia
ls e
xpre
ss t
he
len
gth
an
d w
idth
of
the
pool
?x
�3
and
x�
2
Pra
ctic
e (
Ave
rag
e)
Th
e R
emai
nd
er a
nd
Fac
tor T
heo
rem
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-4
7-4
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
An
swer
s
Answers (Lesson 7-4)
Readin
g t
o L
earn
Math
em
ati
csT
he
Rem
ain
der
an
d F
acto
r Th
eore
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-4
7-4
©G
lenc
oe/M
cGra
w-H
ill39
7G
lenc
oe A
lgeb
ra 2
Lesson 7-4
Pre-
Act
ivit
yH
ow c
an y
ou u
se t
he
Rem
ain
der
Th
eore
m t
o ev
alu
ate
pol
ynom
ials
?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-4
at
the
top
of p
age
365
in y
our
text
book
.
Sh
ow h
ow y
ou w
ould
use
th
e m
odel
in
th
e in
trod
uct
ion
to
esti
mat
e th
en
um
ber
of i
nte
rnat
ion
al t
rave
lers
(in
mil
lion
s) t
o th
e U
nit
ed S
tate
s in
th
eye
ar 2
000.
(Sh
ow h
ow y
ou w
ould
su
bsti
tute
nu
mbe
rs,b
ut
do n
ot a
ctu
ally
calc
ula
te t
he
resu
lt.)
Sam
ple
an
swer
:0.
02(1
4)3
�0.
6(14
)2�
6(14
) �
25.9
Rea
din
g t
he
Less
on
1.C
onsi
der
the
foll
owin
g sy
nth
etic
div
isio
n.
13
2�
64
35
�1
35
�1
3
a.U
sin
g th
e di
visi
on s
ymbo
l �
,wri
te t
he
divi
sion
pro
blem
th
at i
s re
pres
ente
d by
th
issy
nth
etic
div
isio
n.(
Do
not
in
clu
de t
he
answ
er.)
(3x3
�2x
2�
6x�
4) �
(x�
1)
b.
Iden
tify
eac
h o
f th
e fo
llow
ing
for
this
div
isio
n.
divi
den
ddi
viso
r
quot
ien
tre
mai
nde
r
c.If
f(x
) �
3x3
�2x
2�
6x�
4,w
hat
is
f(1)
?3
2.C
onsi
der
the
foll
owin
g sy
nth
etic
div
isio
n.
�3
10
027
�3
9�
271
�3
90
a.T
his
div
isio
n s
how
s th
at
is a
fac
tor
of
.
b.
Th
e di
visi
on s
how
s th
at
is a
zer
o of
th
e po
lyn
omia
l fu
nct
ion
f(x)
�.
c.T
he
divi
sion
sh
ows
that
th
e po
int
is o
n t
he
grap
h o
f th
e po
lyn
omia
l
fun
ctio
n f
(x)
�.
Hel
pin
g Y
ou
Rem
emb
er
3.T
hin
k of
a m
nem
onic
for
rem
embe
rin
g th
e se
nte
nce
,“D
ivid
end
equ
als
quot
ien
t ti
mes
divi
sor
plu
s re
mai
nde
r.”S
amp
le a
nsw
er:
Def
init
ely
ever
y q
uie
t te
ach
er d
eser
ves
pro
per
rew
ard
s.
x3�
27(�
3,0)
x3�
27�
3
x3�
27x
� 3
33x
3�
5x�
1
x �
13x
3�
2x2
�6x
�4
©G
lenc
oe/M
cGra
w-H
ill39
8G
lenc
oe A
lgeb
ra 2
Usi
ng
Max
imu
m V
alu
esM
any
tim
es m
axim
um
sol
uti
ons
are
nee
ded
for
diff
eren
t si
tuat
ion
s.F
or
inst
ance
,wh
at i
s th
e ar
ea o
f th
e la
rges
t re
ctan
gula
r fi
eld
that
can
be
encl
osed
w
ith
200
0 fe
et o
f fe
nci
ng?
Let
xan
d y
den
ote
the
len
gth
an
d w
idth
of
th
e fi
eld,
resp
ecti
vely
.
Per
imet
er:2
x�
2y�
2000
→y
�10
00 �
xA
rea:
A�
xy�
x(10
00 �
x) �
�x2
�10
00x
Th
is p
robl
em i
s eq
uiv
alen
t to
fin
din
g
the
hig
hes
t po
int
on t
he
grap
h o
f A
(x)
��
x2�
1000
xsh
own
on
th
e ri
ght.
Com
plet
e th
e sq
uar
e fo
r �
x2�
1000
x.
A�
�(x
2�
1000
x�
5002
) �
5002
��
(x�
500)
2�
5002
Bec
ause
th
e te
rm �
(x�
500)
2is
eit
her
n
egat
ive
or 0
,th
e gr
eate
st v
alu
e of
Ais
500
2 .T
he
max
imu
m a
rea
encl
osed
is
5002
or 2
50,0
00 s
quar
e fe
et.
Sol
ve e
ach
pro
ble
m.
1.F
ind
the
area
of
the
larg
est
rect
angu
lar
gard
en t
hat
can
be
encl
osed
by
300
feet
of
fen
ce.
5625
ft2
2.A
far
mer
wil
l m
ake
a re
ctan
gula
r pe
n w
ith
100
fee
t of
fen
ce u
sin
g pa
rt
of h
is b
arn
for
on
e si
de o
f th
e pe
n.W
hat
is
the
larg
est
area
he
can
en
clos
e?
1250
ft2
3.A
n a
rea
alon
g a
stra
igh
t st
one
wal
l is
to
be f
ence
d.T
her
e ar
e 60
0 m
eter
s of
fen
cin
g av
aila
ble.
Wh
at i
s th
e gr
eate
st r
ecta
ngu
lar
area
th
at c
an b
e en
clos
ed?
45,0
00 m
2
A
xO
1000
x
y
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-4
7-4
© Glencoe/McGraw-Hill A14 Glencoe Algebra 2
Answers (Lesson 7-5)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Ro
ots
an
d Z
ero
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-5
7-5
©G
lenc
oe/M
cGra
w-H
ill39
9G
lenc
oe A
lgeb
ra 2
Lesson 7-5
Typ
es o
f R
oo
tsT
he f
ollo
win
g st
atem
ents
are
equ
ival
ent
for
any
poly
nom
ial
func
tion
f(x
).•
cis
a z
ero
of t
he
poly
nom
ial
fun
ctio
n f
(x).
•(x
�c)
is
a fa
ctor
of
the
poly
nom
ial
f(x)
.•
cis
a r
oot
or s
olu
tion
of
the
poly
nom
ial
equ
atio
n f
(x)
�0.
If c
is r
eal,
then
(c,
0) i
s an
in
terc
ept
of t
he
grap
h o
f f(
x).
Fu
nd
amen
tal
Eve
ry p
olyn
omia
l equ
atio
n w
ith d
egre
e gr
eate
r th
an z
ero
has
at le
ast
one
root
in t
he s
etT
heo
rem
of
Alg
ebra
of c
ompl
ex n
umbe
rs.
Co
rolla
ry t
o t
he
Apo
lyno
mia
l equ
atio
n of
the
for
m P
(x)
�0
of d
egre
e n
with
com
plex
coe
ffici
ents
has
Fu
nd
amen
tal
exac
tly n
root
s in
the
set
of
com
plex
num
bers
.T
heo
rem
of
Alg
ebra
s
If P
(x)
is a
pol
ynom
ial w
ith r
eal c
oeffi
cien
ts w
hose
ter
ms
are
arra
nged
in d
esce
ndin
gpo
wer
s of
the
var
iabl
e,
Des
cart
es’R
ule
•th
e nu
mbe
r of
pos
itive
rea
l zer
os o
f y
�P
(x)
is t
he s
ame
as t
he n
umbe
r of
cha
nges
in
of
Sig
ns
sign
of
the
coef
ficie
nts
of t
he t
erm
s, o
r is
less
tha
n th
is b
y an
eve
n nu
mbe
r, an
d•
the
num
ber
of n
egat
ive
real
zer
os o
f y
�P
(x)
is t
he s
ame
as t
he n
umbe
r of
cha
nges
in
sign
of
the
coef
ficie
nts
of t
he t
erm
s of
P(�
x),
or is
less
tha
n th
is n
umbe
r by
an
even
num
ber.
Sol
ve t
he
equ
atio
n
6x3
�3x
�0
and
sta
te t
he
nu
mb
er a
nd
typ
e of
roo
ts.
6x3
�3x
�0
3x(2
x2�
1) �
0U
se t
he
Zer
o P
rodu
ct P
rope
rty.
3x�
0or
2x2
�1
�0
x�
0or
2x2
��
1
x�
Th
e eq
uat
ion
has
on
e re
al r
oot,
0,
and
two
imag
inar
y ro
ots,
.
i�2�
�2
i�2�
�2
Sta
te t
he
nu
mb
er o
f p
osit
ive
real
zer
os,n
egat
ive
real
zer
os,a
nd
im
agin
ary
zero
s fo
r p
(x)
�4x
4�
3x3
�x2
�2x
�5.
Sin
ce p
(x)
has
deg
ree
4,it
has
4 z
eros
.U
se D
esca
rtes
’ Ru
le o
f S
ign
s to
det
erm
ine
the
num
ber
and
type
of
real
zer
os.S
ince
the
re a
re t
hree
sign
ch
ange
s,th
ere
are
3 or
1 p
osit
ive
real
zer
os.
Fin
d p(
�x)
an
d co
un
t th
e n
um
ber
of c
han
ges
insi
gn f
or i
ts c
oeff
icie
nts
.p(
�x)
�4(
�x)
4�
3(�
x)3
�(�
x)2
�2(
�x)
�5
�4x
4�
3x3
�x2
�2x
�5
Sin
ce t
her
e is
on
e si
gn c
han
ge,t
her
e is
exa
ctly
1n
egat
ive
real
zer
o.
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Sol
ve e
ach
eq
uat
ion
an
d s
tate
th
e n
um
ber
an
d t
ype
of r
oots
.
1.x2
�4x
�21
�0
2.2x
3�
50x
�0
3.12
x3�
100x
�0
3,�
7;2
real
0,�
5;3
real
0,�
;1
real
,2 im
agin
ary
Sta
te t
he
nu
mb
er o
f p
osit
ive
real
zer
os,n
egat
ive
real
zer
os,a
nd
im
agin
ary
zero
sfo
r ea
ch f
un
ctio
n.
4.f(
x) �
3x3
�x2
�8x
�12
1;2
or
0;0
or
2
5.f(
x) �
2x4
�x3
�3x
�7
2 o
r 0;
0;2
or
4
6.f(
x) �
3x5
�x4
�x3
�6x
2�
53
or
1;2
or
0;0,
2,o
r 4
5i�
3��
3
©G
lenc
oe/M
cGra
w-H
ill40
0G
lenc
oe A
lgeb
ra 2
Fin
d Z
ero
s
Co
mp
lex
Co
nju
gat
eS
uppo
se a
and
bar
e re
al n
umbe
rs w
ith b
0.
If
a�
biis
a z
ero
of a
pol
ynom
ial
Th
eore
mfu
nctio
n w
ith r
eal c
oeffi
cien
ts,
then
a�
biis
als
o a
zero
of
the
func
tion.
Fin
d a
ll o
f th
e ze
ros
of f
(x)
�x4
�15
x2�
38x
�60
.S
ince
f(x
) h
as d
egre
e 4,
the
fun
ctio
n h
as 4
zer
os.
f(x)
�x4
�15
x2�
38x
�60
f(�
x) �
x4�
15x2
�38
x�
60S
ince
th
ere
are
3 si
gn c
han
ges
for
the
coef
fici
ents
of
f(x)
,th
e fu
nct
ion
has
3 o
r 1
posi
tive
rea
lze
ros.
Sin
ce t
her
e is
1 s
ign
ch
ange
for
th
e co
effi
cien
ts o
f f(
�x)
,th
e fu
nct
ion
has
1 n
egat
ive
real
zer
o.U
se s
ynth
etic
su
bsti
tuti
on t
o te
st s
ome
poss
ible
zer
os.
21
0�
1538
�60
24
�22
321
2�
1116
�28
31
0�
1538
�60
39
�18
601
3�
620
0S
o 3
is a
zer
o of
th
e po
lyn
omia
l fu
nct
ion
.Now
try
syn
thet
ic s
ubs
titu
tion
aga
in t
o fi
nd
a ze
roof
th
e de
pres
sed
poly
nom
ial.
�2
13
�6
20�
2�
216
11
�8
36
�4
13
�6
20�
44
81
�1
�2
28
�5
13
�6
20�
510
�20
1�
24
0
So
�5
is a
not
her
zer
o.U
se t
he
Qu
adra
tic
For
mu
la o
n t
he
depr
esse
d po
lyn
omia
l x2
�2x
�4
to f
ind
the
oth
er 2
zer
os,1
i �
3�.T
he
fun
ctio
n h
as t
wo
real
zer
os a
t 3
and
�5
and
two
imag
inar
y ze
ros
at 1
i �
3�.
Fin
d a
ll o
f th
e ze
ros
of e
ach
fu
nct
ion
.
1.f(
x) �
x3�
x2�
9x�
9�
1,�
3i2.
f(x)
�x3
�3x
2�
4x�
123,
�2i
3.p(
a) �
a3�
10a2
�34
a�
404,
3 �
i4.
p(x)
�x3
�5x
2�
11x
�15
3,1
�2i
5.f(
x) �
x3�
6x�
206.
f(x)
�x4
�3x
3�
21x2
�75
x�
100
�2,
1 �
3i�
1,4,
�5i
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Ro
ots
an
d Z
ero
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-5
7-5
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A15 Glencoe Algebra 2
An
swer
s
Answers (Lesson 7-5)
Skil
ls P
ract
ice
Ro
ots
an
d Z
ero
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-5
7-5
©G
lenc
oe/M
cGra
w-H
ill40
1G
lenc
oe A
lgeb
ra 2
Lesson 7-5
Sol
ve e
ach
eq
uat
ion
.Sta
te t
he
nu
mb
er a
nd
typ
e of
roo
ts.
1.5x
�12
�0
2.x2
�4x
�40
�0
��1 52 �
;1
real
2 �
6i;
2 im
agin
ary
3.x5
�4x
3�
04.
x4�
625
�0
0,0,
0,2i
,�2i
;3
real
,2 im
agin
ary
5i,5
i,�
5i,�
5i;
4 im
agin
ary
5.4x
2�
4x�
1 �
06.
x5�
81x
�0
;2
real
0,�
3,3,
�3i
,3i;
3 re
al,2
imag
inar
y
Sta
te t
he
pos
sib
le n
um
ber
of
pos
itiv
e re
al z
eros
,neg
ativ
e re
al z
eros
,an
dim
agin
ary
zero
s of
eac
h f
un
ctio
n.
7.g(
x) �
3x3
�4x
2�
17x
�6
8.h
(x)
�4x
3�
12x2
�x
�3
2 o
r 0;
1;2
or
02
or
0;1;
2 o
r 0
9.f(
x) �
x3�
8x2
�2x
�4
10.p
(x)
�x3
�x2
�4x
�6
3 o
r 1;
0;2
or
03
or
1;0;
2 o
r 0
11.q
(x)
�x4
�7x
2�
3x�
912
.f(x
) �
x4�
x3�
5x2
�6x
�1
1;1;
22
or
0;2
or
0;4
or
2 o
r 0
Fin
d a
ll t
he
zero
s of
eac
h f
un
ctio
n.
13.h
(x)
�x3
�5x
2�
5x�
314
.g(x
) �
x3�
6x2
�13
x�
10
3,1
��
2�,1
��
2�2,
2 �
i,2
�i
15.h
(x)
�x3
�4x
2�
x�
616
.q(x
) �
x3�
3x2
�6x
�8
1,�
2,�
32,
�1,
�4
17.g
(x)
�x4
�3x
3�
5x2
�3x
�4
18.f
(x)
�x4
�21
x2�
80
�1,
�1,
1,4
�4,
4,�
�5�,
�5�
Wri
te a
pol
ynom
ial
fun
ctio
n o
f le
ast
deg
ree
wit
h i
nte
gral
coe
ffic
ien
ts t
hat
has
th
egi
ven
zer
os.
19.�
3,�
5,1
20.3
if(
x)
�x
3�
7x2
�7x
�15
f(x
) �
x2
�9
21.�
5 �
i22
.�1,
�3�,
��
3�f(
x)
�x
2�
10x
�26
f(x
) �
x3
�x
2�
3x�
3
23.i
,5i
24.�
1,1,
i�6�
f(x
) �
x4
�26
x2
�25
f(x
) �
x4
�5x
2�
6
1 �
�2�
�2
©G
lenc
oe/M
cGra
w-H
ill40
2G
lenc
oe A
lgeb
ra 2
Sol
ve e
ach
eq
uat
ion
.Sta
te t
he
nu
mb
er a
nd
typ
e of
roo
ts.
1.�
9x�
15 �
02.
x4�
5x2
�4
�0
��5 3� ;
1 re
al�
1,1,
�2,
2;4
real
3.x5
�81
x4.
x3�
x2�
3x�
3 �
0
0,�
3,3,
�3i
,3i;
3 re
al,2
imag
inar
y�
1,�
�3�,
�3�;
3 re
al
5.x3
�6x
�20
�0
6.x4
�x3
�x2
�x
�2
�0
�2,
1 �
3i;
1 re
al,2
imag
inar
y2,
�1,
�i,
i;2
real
,2 im
agin
ary
Sta
te t
he
pos
sib
le n
um
ber
of
pos
itiv
e re
al z
eros
,neg
ativ
e re
al z
eros
,an
dim
agin
ary
zero
s of
eac
h f
un
ctio
n.
7.f(
x) �
4x3
�2x
2�
x�
38.
p(x)
�2x
4�
2x3
�2x
2�
x�
1
2 o
r 0;
1;2
or
03
or
1;1;
2 o
r 0
9.q(
x) �
3x4
�x3
�3x
2�
7x�
510
.h(x
) �
7x4
�3x
3�
2x2
�x
�1
2 o
r 0;
2 o
r 0;
4,2,
or
02
or
0;2
or
0;4,
2,o
r 0
Fin
d a
ll t
he
zero
s of
eac
h f
un
ctio
n.
11.h
(x)
�2x
3�
3x2
�65
x�
8412
.p(x
) �
x3�
3x2
�9x
�7
�7,
�3 2� ,4
1,1
�i�
6�,1
�i�
6�
13.h
(x)
�x3
�7x
2�
17x
�15
14.q
(x)
�x4
�50
x2�
49
3,2
�i,
2 �
i�
i,i,
�7i
,7i
15.g
(x)
�x4
�4x
3�
3x2
�14
x�
816
.f(x
) �
x4�
6x3
�6x
2�
24x
�40
�1,
�1,
2,�
4�
2,2,
3 �
i,3
�i
Wri
te a
pol
ynom
ial
fun
ctio
n o
f le
ast
deg
ree
wit
h i
nte
gral
coe
ffic
ien
ts t
hat
has
th
egi
ven
zer
os.
17.�
5,3i
18.�
2,3
�i
f(x
) �
x3
�5x
2�
9x�
45f(
x)
�x
3�
4x2
�2x
�20
19.�
1,4,
3i20
.2,5
,1 �
if(
x)
�x
4�
3x3
�5x
2�
27x
�36
f(x
) �
x4
�9x
3�
26x
2�
34x
�20
21.C
RA
FTS
Ste
phan
has
a s
et o
f pl
ans
to b
uil
d a
woo
den
box
.He
wan
ts t
o re
duce
th
evo
lum
e of
th
e bo
x to
105
cu
bic
inch
es.H
e w
ould
lik
e to
red
uce
th
e le
ngt
h o
f ea
chdi
men
sion
in
th
e pl
an b
y th
e sa
me
amou
nt.
Th
e pl
ans
call
for
th
e bo
x to
be
10 i
nch
es b
y8
inch
es b
y 6
inch
es.W
rite
an
d so
lve
a po
lyn
omia
l eq
uat
ion
to
fin
d ou
t h
ow m
uch
Ste
phen
sh
ould
tak
e fr
om e
ach
dim
ensi
on.
(10
�x)
(8 �
x)(6
�x)
�10
5;3
in.
Pra
ctic
e (
Ave
rag
e)
Ro
ots
an
d Z
ero
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-5
7-5
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 7-5)
Readin
g t
o L
earn
Math
em
ati
csR
oo
ts a
nd
Zer
os
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-5
7-5
©G
lenc
oe/M
cGra
w-H
ill40
3G
lenc
oe A
lgeb
ra 2
Lesson 7-5
Pre-
Act
ivit
yH
ow c
an t
he
root
s of
an
eq
uat
ion
be
use
d i
n p
har
mac
olog
y?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-5
at
the
top
of p
age
371
in y
our
text
book
.
Usi
ng
the
mod
el g
iven
in
th
e in
trod
uct
ion
,wri
te a
pol
ynom
ial
equ
atio
nw
ith
0 o
n o
ne
side
th
at c
an b
e so
lved
to
fin
d th
e ti
me
or t
imes
at
wh
ich
ther
e is
100
mil
ligr
ams
of m
edic
atio
n i
n a
pat
ien
t’s b
lood
stre
am.
0.5t
4�
3.5t
3�
100t
2�
350t
�10
0 �
0
Rea
din
g t
he
Less
on
1.In
dica
te w
het
her
eac
h s
tate
men
t is
tru
eor
fal
se.
a.E
very
pol
ynom
ial
equ
atio
n o
f de
gree
gre
ater
th
an o
ne
has
at
leas
t on
e ro
ot i
n t
he
set
of r
eal
nu
mbe
rs.
fals
eb
.If
cis
a r
oot
of t
he
poly
nom
ial
equ
atio
n f
(x)
�0,
then
(x
�c)
is
a fa
ctor
of
the
poly
nom
ial
f(x)
.tr
ue
c.If
(x
�c)
is
a fa
ctor
of
the
poly
nom
ial
f(x)
,th
en c
is a
zer
o of
th
e po
lyn
omia
l fu
nct
ion
f.
fals
ed
.A
pol
ynom
ial
fun
ctio
n f
of d
egre
e n
has
exa
ctly
(n
�1)
com
plex
zer
os.
fals
e
2.L
et f
(x)
�x6
�2x
5�
3x4
�4x
3�
5x2
�6x
�7.
a.W
hat
are
th
e po
ssib
le n
um
bers
of
posi
tive
rea
l ze
ros
of f
?5,
3,o
r 1
b.
Wri
te f
(�x)
in
sim
plif
ied
form
(w
ith
no
pare
nth
eses
).x
6�
2x5
�3x
4�
4x3
�5x
2�
6x�
7W
hat
are
th
e po
ssib
le n
um
bers
of
neg
ativ
e re
al z
eros
of
f?1
c.C
ompl
ete
the
foll
owin
g ch
art
to s
how
th
e po
ssib
le c
ombi
nat
ion
s of
pos
itiv
e re
al z
eros
,n
egat
ive
real
zer
os,a
nd
imag
inar
y ze
ros
of t
he
poly
nom
ial
fun
ctio
n f
.
Nu
mb
er o
fN
um
ber
of
Nu
mb
er o
f To
tal N
um
ber
P
osi
tive
Rea
l Zer
os
Neg
ativ
e R
eal Z
ero
sIm
agin
ary
Zer
os
of
Zer
os
51
06
31
26
11
46
Hel
pin
g Y
ou
Rem
emb
er
3.It
is
easi
er t
o re
mem
ber
mat
hem
atic
al c
once
pts
and
resu
lts
if y
ou r
elat
e th
em t
o ea
chot
her
.How
can
th
e C
ompl
ex C
onju
gate
s T
heo
rem
hel
p yo
u r
emem
ber
the
part
of
Des
cart
es’ R
ule
of
Sig
ns
that
say
s,“o
r is
les
s th
an t
his
nu
mbe
r by
an
eve
n n
um
ber.”
Sam
ple
an
swer
:F
or
a p
oly
no
mia
l fu
nct
ion
in w
hic
h t
he
po
lyn
om
ial h
asre
al c
oef
ficie
nts
,im
agin
ary
zero
s co
me
in c
on
jug
ate
pai
rs.T
her
efo
re,t
her
em
ust
be
an e
ven
nu
mb
er o
f im
agin
ary
zero
s.F
or
each
pai
r o
f im
agin
ary
zero
s,th
e n
um
ber
of
po
siti
ve o
r n
egat
ive
zero
s d
ecre
ases
by
2.
©G
lenc
oe/M
cGra
w-H
ill40
4G
lenc
oe A
lgeb
ra 2
Th
e B
isec
tio
n M
eth
od
fo
r A
pp
roxi
mat
ing
Rea
l Zer
os
Th
e b
isec
tion
met
hod
can
be
use
d to
app
roxi
mat
e ze
ros
of p
olyn
omia
l fu
nct
ion
s li
ke f
(x)
�x3
�x2
�3x
�3.
Sin
ce f
(1)
� �
4 an
d f(
2) �
3,th
ere
is a
t le
ast
one
real
zer
o be
twee
n 1
an
d 2.
Th
e m
idpo
int
of t
his
in
terv
al i
s �1
� 22
��
1.5.
Sin
ce f
(1.5
) �
�1.
875,
the
zero
is
betw
een
1.5
an
d 2.
Th
e m
idpo
int
of t
his
in
terv
al i
s �1.
5 2�2
��
1.75
.Sin
ce
f(1.
75)
is a
bout
0.1
72,t
he z
ero
is b
etw
een
1.5
and
1.75
.Th
e m
idpo
int
of t
his
inte
rval
is �1.
5� 2
1.75
��
1.62
5 an
d f(
1.62
5) i
s ab
out
�0.
94.T
he
zero
is
betw
een
1.62
5 an
d 1.
75.T
he
mid
poin
t of
th
is i
nte
rval
is �1.
625
2�1.
75�
�1.
6875
.Sin
ce
f(1.
6875
) is
abo
ut
�0.
41,t
he
zero
is
betw
een
1.6
875
and
1.75
.Th
eref
ore,
the
zero
is
1.7
to t
he
nea
rest
ten
th.
Th
e di
agra
m b
elow
su
mm
ariz
es t
he
resu
lts
obta
ined
by
the
bise
ctio
n m
eth
od.
Usi
ng
the
bis
ecti
on m
eth
od,a
pp
roxi
mat
e to
th
e n
eare
st t
enth
th
e ze
ro b
etw
een
th
e tw
o in
tegr
al v
alu
es o
f x
for
each
fu
nct
ion
.
1.f(
x) �
x3�
4x2
�11
x �
2,f(
0) �
2,f(
1) �
�12
0.2
2.f(
x) �
2x4
�x2
�15
,f(1
) �
�12
,f(2
) �
211.
6
3.f(
x) �
x5�
2x3
�12
,f(1
) �
�13
,f(2
) �
41.
9
4.f(
x) �
4x3
�2x
�7,
f(�
2) �
�21
,f(�
1) �
5�
1.3
5.f(
x) �
3x3
�14
x2�
27x
�12
6,f(
4) �
�14
,f(5
) �
164.
7
11.
52
1.62
51.
75
1.68
75
++
––
––
sign
of f
(x):
valu
e x
:
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-5
7-5
© Glencoe/McGraw-Hill A17 Glencoe Algebra 2
An
swer
s
Answers (Lesson 7-6)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Rat
ion
al Z
ero
Th
eore
m
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-6
7-6
©G
lenc
oe/M
cGra
w-H
ill40
5G
lenc
oe A
lgeb
ra 2
Lesson 7-6
Iden
tify
Rat
ion
al Z
ero
s
Rat
ion
al Z
ero
Le
t f(
x) �
a 0xn
�a 1
xn�
1�
… �
a n�
2x2
�a n
�1x
�an
repr
esen
t a
poly
nom
ial f
unct
ion
Th
eore
mw
ith in
tegr
al c
oeffi
cien
ts.
If �p q�
is a
rat
iona
l num
ber
in s
impl
est
form
and
is a
zer
o of
y�
f(x)
,th
en p
is a
fac
tor
of a
nan
d q
is a
fac
tor
of a
0.
Co
rolla
ry (
Inte
gra
l If
the
coef
ficie
nts
of a
pol
ynom
ial a
re in
tege
rs s
uch
that
a0
�1
and
a n
0, a
ny r
atio
nal
Zer
o T
heo
rem
)ze
ros
of t
he f
unct
ion
mus
t be
fac
tors
of
a n.
Lis
t al
l of
th
e p
ossi
ble
rat
ion
al z
eros
of
each
fu
nct
ion
.
a.f(
x) �
3x4
�2x
2�
6x�
10
If �p q�
is a
rat
ion
al r
oot,
then
pis
a f
acto
r of
�10
an
d q
is a
fac
tor
of 3
.Th
e po
ssib
le v
alu
es
for
par
e
1,
2,
5,an
d
10.T
he
poss
ible
val
ues
for
qar
e
1 an
d
3.S
o al
l of
th
e po
ssib
le r
atio
nal
zer
os a
re �p q�
�
1,
2,
5,
10,
�1 3� ,
�2 3� ,
�5 3� ,an
d
�1 30 �.
b.
q(x
) �
x3�
10x2
�14
x�
36
Sin
ce t
he
coef
fici
ent
of x
3is
1,t
he
poss
ible
rat
ion
al z
eros
mu
st b
e th
e fa
ctor
s of
th
eco
nst
ant
term
�36
.So
the
poss
ible
rat
ion
al z
eros
are
1,
2,
3,
4,
6,
9,
12
,18
,an
d
36.
Lis
t al
l of
th
e p
ossi
ble
rat
ion
al z
eros
of
each
fu
nct
ion
.
1.f(
x) �
x3�
3x2
�x
�8
2.g(
x) �
x5�
7x4
�3x
2�
x�
20
�1,
�2,
�4,
�8
�1,
�2,
�4,
�5,
�10
,�20
3.h
(x)
�x4
�7x
3�
4x2
�x
�49
4.p(
x) �
2x4
�5x
3�
8x2
�3x
�5
�1,
�7,
�49
�1,
�5,
�,�
5.q(
x) �
3x4
�5x
3�
10x
�12
6.r(
x) �
4x5
�2x
�18
�1,
�2,
�3,
�4,
�6,
�12
,�
1,�
2,�
3,�
6,�
9,�
18,
�,�
,��
,�,�
,�,�
,�
7.f(
x) �
x7�
6x5
�3x
4�
x3�
4x2
�12
08.
g(x)
�5x
6�
3x4
�5x
3�
2x2
�15
�1,
�2,
�3,
�4,
�5,
�6,
�8,
�10
,�12
,�
15,�
20,�
24,�
30,�
40,�
60,�
120
�1,
�3,
�5,
�15
,�,�
9.h
(x)
�6x
5�
3x4
�12
x3�
18x2
�9x
�21
10.p
(x)
�2x
7�
3x6
�11
x5�
20x2
�11
�1,
�3,
�7,
�21
,�,�
,�,�
,�
1,�
11,�
,�
�,�
,�,�
7 � 61 � 6
7 � 31 � 3
11 � 21 � 2
21 � 27 � 2
3 � 21 � 2
3 � 51 � 5
9 � 43 � 4
1 � 49 � 2
3 � 21 � 2
4 � 32 � 3
1 � 3
5 � 21 � 2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill40
6G
lenc
oe A
lgeb
ra 2
Fin
d R
atio
nal
Zer
os
Fin
d a
ll o
f th
e ra
tion
al z
eros
of
f(x)
�5x
3�
12x2
�29
x�
12.
Fro
m t
he
coro
llar
y to
th
e F
un
dam
enta
l Th
eore
m o
f Alg
ebra
,we
know
th
at t
her
e ar
e ex
actl
y 3
com
plex
roo
ts.A
ccor
din
g to
Des
cart
es’ R
ule
of
Sig
ns
ther
e ar
e 2
or 0
pos
itiv
e re
al r
oots
an
d 1
neg
ativ
e re
al r
oot.
Th
e po
ssib
le r
atio
nal
zer
os a
re
1,
2,
3,
4,
6,
12,
,
,,
,,
.Mak
e a
tabl
e an
d te
st s
ome
poss
ible
rat
ion
al z
eros
.
Sin
ce f
(1)
�0,
you
kn
ow t
hat
x�
1 is
a z
ero.
Th
e de
pres
sed
poly
nom
ial
is 5
x2�
17x
�12
,wh
ich
can
be
fact
ored
as
(5x
�3)
(x�
4).
By
the
Zer
o P
rodu
ct P
rope
rty,
this
exp
ress
ion
equ
als
0 w
hen
x�
or x
��
4.T
he
rati
onal
zer
os o
f th
is f
un
ctio
n a
re 1
,,a
nd
�4.
Fin
d a
ll o
f th
e ze
ros
of f
(x)
�8x
4�
2x3
�5x
2�
2x�
3.T
her
e ar
e 4
com
plex
roo
ts,w
ith
1 p
osit
ive
real
roo
t an
d 3
or 1
neg
ativ
e re
al r
oots
.Th
e po
ssib
le r
atio
nal
zer
os a
re
1,
3,
,,
,,
,an
d
.3 � 8
3 � 43 � 2
1 � 81 � 4
1 � 2
3 � 5
3 � 5
�p q�5
12�
2912
15
17�
120
12 � 56 � 5
4 � 53 � 5
2 � 51 � 5
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Rat
ion
al Z
ero
Th
eore
m
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-6
7-6
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Mak
e a
tabl
e an
d te
st s
ome
poss
ible
val
ues
.
Sin
ce f�
��0,
we
know
tha
t x
�
is a
roo
t.
1 � 21 � 2
�p q�8
25
2�
3
18
1015
1714
28
1841
8416
5
�1 2�8
68
60
The
dep
ress
ed p
olyn
omia
l is
8x3
�6x
2�
8x�
6.T
ry s
ynth
etic
su
bsti
tuti
on a
gain
.An
y re
mai
nin
gra
tion
al r
oots
mu
st b
e n
egat
ive.
x�
��3 4�
is a
noth
er r
atio
nal r
oot.
The
dep
ress
ed p
olyn
omia
l is
8x2
�8
�0,
whi
ch h
as r
oots
i.
�p q�8
68
6
��1 4�
84
74�
1 4�
��3 4�
80
80
Th
e ze
ros
of t
his
fu
nct
ion
are
�1 2� ,�
�3 4� ,an
d
i.
Fin
d a
ll o
f th
e ra
tion
al z
eros
of
each
fu
nct
ion
.
1.f(
x) �
x3�
4x2
�25
x�
28�
1,4,
�7
2.f(
x) �
x3�
6x2
�4x
�24
�6
Fin
d a
ll o
f th
e ze
ros
of e
ach
fu
nct
ion
.
3.f(
x) �
x4�
2x3
�11
x2�
8x�
604.
f(x)
�4x
4�
5x3
�30
x2�
45x
�54
3,�
5,�
2i,�
2,�
3i3 � 4
© Glencoe/McGraw-Hill A18 Glencoe Algebra 2
Answers (Lesson 7-6)
Skil
ls P
ract
ice
Rat
ion
al Z
ero
Th
eore
m
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-6
7-6
©G
lenc
oe/M
cGra
w-H
ill40
7G
lenc
oe A
lgeb
ra 2
Lesson 7-6
Lis
t al
l of
th
e p
ossi
ble
rat
ion
al z
eros
of
each
fu
nct
ion
.
1.n
(x)
�x2
�5x
�3
2.h
(x)
�x2
�2x
�5
�1,
�3
�1,
�5
3.w
(x)
�x2
�5x
�12
4.f(
x) �
2x2
�5x
�3
�1,
�2,
�3,
�4,
�6,
�12
��1 2� ,
��3 2� ,
�1,
�3
5.q(
x) �
6x3
�x2
�x
�2
6.g(
x) �
9x4
�3x
3�
3x2
�x
�27
��1 6� ,
��1 3� ,
��1 2� ,
��2 3� ,
�1,
�2
��1 9� ,
��1 3� ,
�1,
�3,
�9,
�27
Fin
d a
ll o
f th
e ra
tion
al z
eros
of
each
fu
nct
ion
.
7.f(
x) �
x3�
2x2
�5x
�4
�0
8.g(
x) �
x3�
3x2
�4x
�12
1�
2,2,
3
9.p(
x) �
x3�
x2�
x�
110
.z(x
) �
x3�
4x2
�6x
�4
12
11.h
(x)
�x3
�x2
�4x
�4
12.g
(x)
�3x
3�
9x2
�10
x�
8
14
13.g
(x)
�2x
3�
7x2
�7x
�12
14.h
(x)
�2x
3�
5x2
�4x
�3
�4,
�1,
�3 2��
1,�1 2� ,
3
15.p
(x)
�3x
3�
5x2
�14
x�
4 �
016
.q(x
) �
3x3
�2x
2�
27x
�18
��1 3�
��2 3�
17.q
(x)
�3x
3�
7x2
�4
18.f
(x)
�x4
�2x
3�
13x2
�14
x�
24
��2 3� ,
1,2
�3,
�1,
2,4
19.p
(x)
�x4
�5x
3�
9x2
�25
x�
7020
.n(x
) �
16x4
�32
x3�
13x2
�29
x�
6
�2,
7�
1,�1 4� ,
�3 4� ,2
Fin
d a
ll o
f th
e ze
ros
of e
ach
fu
nct
ion
.
21.f
(x)
�x3
�5x
2�
11x
�15
22.q
(x)
�x3
�10
x2�
18x
�4
�3,
�1
�2i
,�1
�2i
2,4
��
14�,4
��
14�
23.m
(x)
�6x
4�
17x3
�8x
2�
8x�
324
.g(x
) �
x4�
4x3
�5x
2�
4x�
4
�1 3� ,�3 2� ,
,�
2,�
2,�
i,i
1 �
�5�
�2
1 �
�5�
�2
©G
lenc
oe/M
cGra
w-H
ill40
8G
lenc
oe A
lgeb
ra 2
Lis
t al
l of
th
e p
ossi
ble
rat
ion
al z
eros
of
each
fu
nct
ion
.
1.h
(x)
�x3
� 5
x2�
2x�
122.
s(x)
�x4
� 8
x3�
7x�
14
�1,
�2,
�3,
�4,
�6,
�12
�1,
�2,
�7,
�14
3.f(
x) �
3x5
�5x
2�
x�
64.
p(x)
�3x
2�
x�
7
��1 3� ,
��2 3� ,
�1,
�2,
�3,
�6
��1 3� ,
��7 3� ,
�1,
�7
5.g(
x) �
5x3
�x2
�x
�8
6.q(
x) �
6x5
�x3
�3
��1 5� ,
��2 5� ,
��4 5� ,
��8 5� ,
�1,
�2,
�4,
�8
��1 6� ,
��1 3� ,
��1 2� ,
��3 2� ,
�1,
�3
Fin
d a
ll o
f th
e ra
tion
al z
eros
of
each
fu
nct
ion
.
7.q(
x) �
x3�
3x2
�6x
�8
�0
�4,
�1,
28.
v(x)
�x3
�9x
2�
27x
�27
3
9.c(
x) �
x3�
x2�
8x�
12�
3,2
10.f
(x)
�x4
�49
x20,
�7,
7
11.h
(x)
�x3
�7x
2�
17x
�15
312
.b(x
) �
x3�
6x�
20�
2
13.f
(x)
�x3
�6x
2�
4x�
246
14.g
(x)
�2x
3�
3x2
�4x
�4
�2
15.h
(x)
�2x
3�
7x2
�21
x�
54 �
0�3,
2,�9 2�
16.z
(x)
�x4
�3x
3�
5x2
�27
x�
36�
1,4
17.d
(x)
�x4
�x3
�16
no
rat
ion
al z
ero
s18
.n(x
) �
x4�
2x3
�3
�1
19.p
(x)
�2x
4�
7x3
�4x
2�
7x�
620
.q(x
) �
6x4
�29
x3�
40x2
�7x
�12
�1,
1,�3 2� ,
2�
�3 2� ,�
�4 3�
Fin
d a
ll o
f th
e ze
ros
of e
ach
fu
nct
ion
.
21.f
(x)
�2x
4�
7x3
�2x
2�
19x
�12
22.q
(x)
�x4
�4x
3�
x2�
16x
�20
�1,
�3,
,�
2,2,
2 �
i,2
�i
23.h
(x)
�x6
�8x
324
.g(x
) �
x6�
1�
1,1,
,
0,2,
�1
�i�
3�,�
1 �
i�3�
,,
25.T
RA
VEL
Th
e h
eigh
t of
a b
ox t
hat
Joa
n i
s sh
ippi
ng
is 3
in
ches
les
s th
an t
he
wid
th o
f th
ebo
x.T
he
len
gth
is
2 in
ches
mor
e th
an t
wic
e th
e w
idth
.Th
e vo
lum
e of
th
e bo
x is
154
0 in
3 .W
hat
are
th
e di
men
sion
s of
th
e bo
x?22
in.b
y 10
in.b
y 7
in.
26.G
EOM
ETRY
The
hei
ght
of a
squ
are
pyra
mid
is 3
met
ers
shor
ter
than
the
sid
e of
its
base
.If
th
e vo
lum
e of
th
e py
ram
id i
s 43
2 m
3 ,h
ow t
all
is i
t? U
se t
he
form
ula
V�
�1 3� Bh
.9
m
1 �
i�3�
�� 2
1 �
i�3�
�� 2
�1
�i�
3��
� 2
�1
�i�
3��
� 2
1 �
�33�
�� 4
1 �
�33�
�� 4
Pra
ctic
e (
Ave
rag
e)
Rat
ion
al Z
ero
Th
eore
m
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-6
7-6
© Glencoe/McGraw-Hill A19 Glencoe Algebra 2
An
swer
s
Answers (Lesson 7-6)
Readin
g t
o L
earn
Math
em
ati
csR
atio
nal
Zer
o T
heo
rem
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-6
7-6
©G
lenc
oe/M
cGra
w-H
ill40
9G
lenc
oe A
lgeb
ra 2
Lesson 7-6
Pre-
Act
ivit
yH
ow c
an t
he
Rat
ion
al Z
ero
Th
eore
m s
olve
pro
ble
ms
invo
lvin
g la
rge
nu
mb
ers?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-6
at
the
top
of p
age
378
in y
our
text
book
.
Rew
rite
th
e po
lyn
omia
l eq
uat
ion
w(w
�8)
(w�
5) �
2772
in
th
e fo
rm
f(x)
�0,
wh
ere
f(x)
is
a po
lyn
omia
l w
ritt
en i
n d
esce
ndi
ng
pow
ers
of x
.w
3�
3w2
�40
w�
2772
�0
Rea
din
g t
he
Less
on
1.F
or e
ach
of
the
foll
owin
g po
lyn
omia
l fu
nct
ion
s,li
st a
ll t
he
poss
ible
val
ues
of
p,al
l th
e po
ssib
le v
alu
es o
f q,
and
all
the
poss
ible
rat
ion
al z
eros
�p q�.
a.f(
x) �
x3�
2x2
�11
x�
12
poss
ible
val
ues
of
p:�
1,�
2,�
3,�
4,�
6,�
12
poss
ible
val
ues
of
q:�
1
poss
ible
val
ues
of
�p q�:�
1,�
2,�
3,�
4,�
6,�
12
b.
f(x)
�2x
4�
9x3
�23
x2�
81x
�45
poss
ible
val
ues
of
p:�
1,�
3,�
5,�
9,�
15,�
45
poss
ible
val
ues
of
q:�
1,�
2
poss
ible
val
ues
of
�p q�:�
1,�
3,�
5,�
9,�
15,�
45,�
�1 2� ,�
�3 2� ,�
�5 2� ,�
�9 2� ,�
�1 25 �,�
�4 25 �
2.E
xpla
in i
n yo
ur o
wn
wor
ds h
ow D
esca
rtes
’ Rul
e of
Sig
ns,t
he R
atio
nal
Zero
The
orem
,and
syn
thet
ic d
ivis
ion
can
be
use
d to
geth
er t
o fi
nd
all
of t
he
rati
onal
zer
os o
f a
poly
nom
ial
fun
ctio
n w
ith
in
tege
r co
effi
cien
ts.
Sam
ple
an
swer
:U
se D
esca
rtes
’Ru
le t
o f
ind
th
e p
oss
ible
nu
mb
ers
of
po
siti
ve a
nd
neg
ativ
e re
al z
ero
s.U
se t
he
Rat
ion
al Z
ero
Th
eore
m t
o li
st a
llp
oss
ible
rat
ion
al z
ero
s.U
se s
ynth
etic
div
isio
n t
o t
est
wh
ich
of
the
nu
mb
ers
on
th
e lis
t o
f p
oss
ible
rat
ion
al z
ero
s ar
e ac
tual
ly z
ero
s o
f th
ep
oly
no
mia
l fu
nct
ion
.(D
esca
rtes
’Ru
le m
ay h
elp
yo
u t
o li
mit
th
ep
oss
ibili
ties
.)
Hel
pin
g Y
ou
Rem
emb
er
3.S
ome
stud
ents
hav
e tr
oubl
e re
mem
beri
ng w
hich
num
bers
go
in t
he n
umer
ator
s an
d w
hich
go i
n t
he
den
omin
ator
s w
hen
for
min
g a
list
of
poss
ible
rat
ion
al z
eros
of
a po
lyn
omia
lfu
nct
ion
.How
can
you
use
th
e li
nea
r po
lyn
omia
l eq
uat
ion
ax
�b
�0,
wh
ere
aan
d b
are
non
zero
in
tege
rs,t
o re
mem
ber
this
?S
amp
le a
nsw
er:T
he
solu
tio
n o
f th
e eq
uat
ion
is �
�b a� .T
he
nu
mer
ato
r b
is a
fac
tor
of
the
con
stan
t te
rm in
ax
�b
.Th
e d
eno
min
ato
r a
is a
fac
tor
of
the
lead
ing
co
effi
cien
t in
ax
�b
.
©G
lenc
oe/M
cGra
w-H
ill41
0G
lenc
oe A
lgeb
ra 2
Infi
nit
e C
on
tin
ued
Fra
ctio
ns
Som
e in
fin
ite
expr
essi
ons
are
actu
ally
equ
al t
o re
aln
um
bers
! Th
e in
fin
ite
con
tin
ued
fra
ctio
n a
t th
e ri
ght
ison
e ex
ampl
e.
If y
ou u
se x
to s
tan
d fo
r th
e in
fin
ite
frac
tion
,th
en t
he
enti
re d
enom
inat
or o
f th
e fi
rst
frac
tion
on
th
e ri
ght
isal
so e
qual
to
x.T
his
obs
erva
tion
lea
ds t
o th
e fo
llow
ing
equ
atio
n:
x�
1 �
�1 x�
Wri
te a
dec
imal
for
eac
h c
onti
nu
ed f
ract
ion
.
1.1
��1 1�
22.
1 �
1.5
3.1
�1.
66
4.1
�1.
65.
1 �
1.62
5
6.T
he
mor
e te
rms
you
add
to
the
frac
tion
s ab
ove,
the
clos
er t
hei
r va
lue
appr
oach
es t
he
valu
e of
th
e in
fin
ite
con
tin
ued
fra
ctio
n.W
hat
val
ue
do t
he
frac
tion
s se
em t
o be
app
roac
hin
g?ab
ou
t 1.
6
7.R
ewri
te x
�1
��1 x�
as a
qu
adra
tic
equ
atio
n a
nd
solv
e fo
r x.
x2
�x
�1
�0;
x�
;x
�1.
618
or
�0.
618
(Th
e p
osi
tive
ro
ot
is t
he
valu
e o
f th
e in
fin
ite
frac
tio
n,
bec
ause
th
e o
rig
inal
fra
ctio
n is
cle
arly
no
t n
egat
ive.
)
8.F
ind
the
valu
e of
th
e fo
llow
ing
infi
nit
e co
nti
nu
ed f
ract
ion
.
3 �
x�
3 �
�1 x� ;x
�o
r ab
ou
t 3.
303
��
13��
� 2
1
3 �
1
3 �
1
3 �
13
�…
1 �
�5�
�2
1
1 �
1
1 �
1
1 �
1
1 �
1 1
1
1 �
1
1 �
1
1 �
1 1
61
1 �
1
1 �
1 1
1
1 �
1 1
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-6
7-6
x�
1 �
1
1 �
1
1 �
1
1 �
11
�…
© Glencoe/McGraw-Hill A20 Glencoe Algebra 2
Answers (Lesson 7-7)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Op
erat
ion
s o
n F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-7
7-7
©G
lenc
oe/M
cGra
w-H
ill41
1G
lenc
oe A
lgeb
ra 2
Lesson 7-7
Ari
thm
etic
Op
erat
ion
s
Sum
(f�
g)(x
) �
f(x)
�g
(x)
Diff
eren
ce(f
�g)
(x)
�f(
x) �
g(x
)O
per
atio
ns
wit
h F
un
ctio
ns
Pro
duct
(f�
g)(x
) �
f(x)
�g
(x)
Quo
tient
��(x
) �
, g
(x)
0
Fin
d (
f�
g)(x
),(f
�g)
(x),
(f
g)(x
),an
d �
�(x)
for
f(x)
�x2
�3x
�4
and
g(x
) �
3x�
2.(f
�g)
(x)
�f(
x) �
g(x)
Add
ition
of
func
tions
�(x
2�
3x�
4) �
(3x
�2)
f(x)
�x2
�3x
�4,
g(x
) �
3x�
2
�x2
�6x
�6
Sim
plify
.
(f�
g)(x
) �
f(x)
�g(
x)S
ubtr
actio
n of
fun
ctio
ns
�(x
2�
3x�
4) �
(3x
�2)
f(x)
�x2
�3x
�4,
g(x
) �
3x�
2
�x2
�2
Sim
plify
.
(f�
g)(x
)�
f(x)
�g(
x)M
ultip
licat
ion
of f
unct
ions
�(x
2�
3x�
4)(3
x�
2)f(
x) �
x2�
3x�
4, g
(x)
�3x
�2
�x2
(3x
�2)
�3x
(3x
�2)
�4(
3x�
2)D
istr
ibut
ive
Pro
pert
y
�3x
3�
2x2
�9x
2�
6x�
12x
�8
Dis
trib
utiv
e P
rope
rty
�3x
3�
7x2
�18
x�
8S
impl
ify.
��(x
)�
Div
isio
n of
fun
ctio
ns
�,x
�2 3�
f(x)
�x2
�3x
�4
and
g(x
) �
3x�
2
Fin
d (
f�
g)(x
),(f
�g)
(x),
(f
g)(x
),an
d �
�(x)
for
each
f(x
) an
d g
(x).
1.f(
x) �
8x�
3;g(
x) �
4x�
52.
f(x)
�x2
�x
�6;
g(x)
�x
�2
12x
�2;
4x�
8;32
x2
�28
x�
15;
x2
�2x
�8;
x2
�4;
,x
�x
3�
x2
�8x
�12
;x
�3,
x
2
3.f(
x) �
3x2
�x
�5;
g(x)
�2x
�3
4.f(
x) �
2x�
1;g(
x) �
3x2
�11
x�
4
3x2
�x
�2;
3x2
�3x
�8;
3x2
�13
x�
5;�
3x2
�9x
�3;
6x3
�11
x2
�13
x�
15;
6x3
�19
x2
�19
x�
4;
,x
,x
,�4
5.f(
x) �
x2�
1;g(
x) �
x2
�1
�;
x2
�1
�;
x�
1;x
3�
x2
�x
�1,
x
�1
1� x
�1
1� x
�1
1� x
�1
1 � 32x
�1
��
(3x
�1)
(x�
4)3 � 2
3x2
�x
�5
��
2x�
3
5 � 48x
�3
� 4x�
5
f � g
x2�
3x�
4�
�3x
�2
f(x)
� g(x)
f � g
f � g
f(x)
� g(x
)f � g
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill41
2G
lenc
oe A
lgeb
ra 2
Co
mp
osi
tio
n o
f Fu
nct
ion
s
Co
mp
osi
tio
n
Sup
pose
fan
d g
are
func
tions
suc
h th
at t
he r
ange
of
gis
a s
ubse
t of
the
dom
ain
of f
.o
f F
un
ctio
ns
The
n th
e co
mpo
site
fun
ctio
n f
�g
can
be d
escr
ibed
by
the
equa
tion
[f�
g](
x) �
f[g
(x)]
.
For
f�
{(1,
2),(
3,3)
,(2,
4),(
4,1)
} an
d g
�{(
1,3)
,(3,
4),(
2,2)
,(4,
1)},
fin
d f
�g
and
g�
fif
th
ey e
xist
.f[
g(1)
] �
f(3)
�3
f[g(
2)]
�f(
2) �
4f[
g(3)
] �
f(4)
�1
f[g(
4)]
�f(
1) �
2f
�g
�{(
1,3)
,(2,
4),(
3,1)
,(4,
2)}
g[f(
1)]
�g(
2) �
2g[
f(2)
] �
g(4)
�1
g[f(
3)]
�g(
3) �
4g[
f(4)
] �
g(1)
�3
g�
f�
{(1,
2),(
2,1)
,(3,
4),(
4,3)
}
Fin
d [
g�
h](
x) a
nd
[h
�g]
(x)
for
g(x)
�3x
�4
and
h(x
) �
x2�
1.[g
�h
](x)
�g[
h(x
)][h
�g]
(x)
�h
[g(x
)]�
g(x2
�1)
�h
(3x
�4)
�3(
x2�
1) �
4�
(3x
�4)
2�
1�
3x2
�7
�9x
2�
24x
�16
�1
�9x
2�
24x
�15
For
eac
h s
et o
f or
der
ed p
airs
,fin
d f
�g
and
g�
fif
th
ey e
xist
.
1.f
�{(
�1,
2),(
5,6)
,(0,
9)},
2.f
�{(
5,�
2),(
9,8)
,(�
4,3)
,(0,
4)},
g�
{(6,
0),(
2,�
1),(
9,5)
}g
�{(
3,7)
,(�
2,6)
,(4,
�2)
,(8,
10)}
f�
g�
{(2,
2),(
6,9)
,(9,
6)};
f�
gd
oes
no
t ex
ist;
g�
f�
{(�
1,�
1),(
0,5)
,(5,
0)}
g�
f�
{(�
4,7)
,(0,
�2)
,(5,
6),(
9,10
)}
Fin
d [
f�
g](x
) an
d [
g�
f](x
).
3.f(
x) �
2x�
7;g(
x) �
�5x
�1
4.f(
x) �
x2�
1;g(
x) �
�4x
2
[f�
g](
x)
��
10x
�5,
[f�
g](
x)
�16
x4
�1,
[g�
f](x
) �
�10
x�
36[g
�f]
(x)
��
4x4
�8x
2�
4
5.f(
x) �
x2�
2x;g
(x)
�x
�9
6.f(
x) �
5x�
4;g(
x) �
3 �
x[f
�g
](x
) �
x2
�16
x�
63,
[f�
g](
x)
�19
�5x
,[g
�f]
(x)
�x
2�
2x�
9[g
�f]
(x)
��
1 �
5x
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Op
erat
ion
s o
n F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-7
7-7
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A21 Glencoe Algebra 2
An
swer
s
Answers (Lesson 7-7)
Skil
ls P
ract
ice
Op
erat
ion
s o
n F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-7
7-7
©G
lenc
oe/M
cGra
w-H
ill41
3G
lenc
oe A
lgeb
ra 2
Lesson 7-7
Fin
d (
f�
g)(x
),(f
�g)
(x),
(f
g)(x
),an
d �
�(x)
for
each
f(x
) an
d g
(x).
1.f(
x) �
x�
52x
�1;
9;2.
f(x)
�3x
�1
5x�
2;x
�4;
6x2
�7x
�3;
g(x)
�x
�4
�x x� �
5 4�
,x
4g(
x) �
2x�
3�3 2x x
� �1 3
�,x
�3 2�
3.f(
x) �
x2x
2�
x�
4;x2
�x
�4;
4.f(
x) �
3x2�3x
3 x�5
�,x
0;
�3x3 x�
5�
,x
0;
g(x)
�4
�x
4x2
�x3
;,x
4
g(x)
��5 x�
15x,
x
0;�3 5x3 �
,x
0
For
eac
h s
et o
f or
der
ed p
airs
,fin
d f
�g
and
g�
fif
th
ey e
xist
.
5.f
�{(
0,0)
,(4,
�2)
}6.
f�
{(0,
�3)
,(1,
2),(
2,2)
}g
�{(
0,4)
,(�
2,0)
,(5,
0)}
g�
{(�
3,1)
,(2,
0)}
{(0,
�2)
,(�
2,0)
,(5,
0)};
{(�
3,2)
,(2,
�3)
};{(
0,4)
,(4,
0)}
{(0,
1),(
1,0)
,(2,
0)}
7.f
�{(
�4,
3),(
�1,
1),(
2,2)
}8.
f�
{(6,
6),(
�3,
�3)
,(1,
3)}
g�
{(1,
�4)
,(2,
�1)
,(3,
�1)
}g
�{(
�3,
6),(
3,6)
,(6,
�3)
}{(
1,3)
,(2,
1),(
3,1)
};{(
�3,
6),(
3,6)
,(6,
�3)
};{(
�4,
�1)
,(�
1,�
4),(
2,�
1)}
{(6,
�3)
,(�
3,6)
,(1,
6)}
Fin
d [
g�
h](
x) a
nd
[h
�g]
(x).
9.g(
x) �
2x2x
�4;
2x�
210
.g(x
) �
�3x
�12
x�
3;�
12x
�1
h(x
) �
x�
2h
(x)
�4x
�1
11.g
(x)
�x
�6
x;x
12.g
(x)
�x
�3
x2
�3;
x2
�6x
�9
h(x
) �
x�
6h
(x)
�x2
13.g
(x)
�5x
5x2
�5x
�5;
14.g
(x)
�x
�2
2x2
�1;
2x2
�8x
�5
h(x
) �
x2�
x�
125
x2
�5x
�1
h(x
) �
2x2
�3
If f
(x)
�3x
,g(x
) �
x�
4,an
d h
(x)
�x2
�1,
fin
d e
ach
val
ue.
15.f
[g(1
)]15
16.g
[h(0
)]3
17.g
[f(�
1)]
1
18.h
[f(5
)]22
419
.g[h
(�3)
]12
20.h
[f(1
0)]
899
21.f
[h(8
)]18
922
.[f
�(h
�g)
](1)
7223
.[f
�(g
�h
)](�
2)21
x2
� 4 �
x
f � g
x2
�x
�20
;
©G
lenc
oe/M
cGra
w-H
ill41
4G
lenc
oe A
lgeb
ra 2
Fin
d (
f�
g)(x
),(f
�g)
(x),
(f
g)(x
),an
d �� gf � �(
x) f
or e
ach
f(x
) an
d g
(x).
1.f(
x) �
2x�
12.
f(x)
�8x
23.
f(x)
�x2
�7x
�12
g(x)
�x
�3
g(x)
�g(
x) �
x2�
9
3x�
2;x
�4;
�8x4 x
� 21
�,x
0;
2x2
�7x
�3;
7x�
21;
2x2
�5x
�3;
�8x4 x2�
1�
,x
0;x
4�
7x3
�3x
2�
63x
�10
8;
�2 xx ��31
�,x
3
8,x
0;
8x4 ,
x
0�x x
� �4 3
�,x
�
3
For
eac
h s
et o
f or
der
ed p
airs
,fin
d f
�g
and
g�
fif
th
ey e
xist
.
4.f
�{(
�9,
�1)
,(�
1,0)
,(3,
4)}
5.f
�{(
�4,
3),(
0,�
2),(
1,�
2)}
g�
{(0,
�9)
,(�
1,3)
,(4,
�1)
}g
�{(
�2,
0),(
3,1)
}{(
0,�
1),(
�1,
4),(
4,0)
};{(
�2,
�2)
,(3,
�2)
};{(
�9,
3),(
�1,
�9)
,(3,
�1)
}{(
�4,
1),(
0,0)
,(1,
0)}
6.f
�{(
�4,
�5)
,(0,
3),(
1,6)
}7.
f�
{(0,
�3)
,(1,
�3)
,(6,
8)}
g�
{(6,
1),(
�5,
0),(
3,�
4)}
g�
{(8,
2),(
�3,
0),(
�3,
1)}
{(6,
6),(
�5,
3),(
3,�
5)};
do
es n
ot
exis
t;{(
�4,
0),(
0,�
4),(
1,1)
}{(
0,0)
,(1,
0),(
6,2)
}
Fin
d [
g�
h](
x) a
nd
[h
�g]
(x).
8.g(
x) �
3x9.
g(x)
��
8x10
.g(x
) �
x�
6h
(x)
�x
�4
h(x
) �
2x�
3h
(x)
�3x
23x
2�
6;3x
�12
;3x
�4
�16
x�
24;
�16
x�
33x
2�
36x
�10
8
11.g
(x)
�x
�3
12.g
(x)
��
2x13
.g(x
) �
x�
2h
(x)
�2x
2h
(x)
�x2
�3x
�2
h(x
) �
3x2
�1
2x2
�3;
�2x
2�
6x�
4;3x
2�
1;2x
2�
12x
�18
4x2
�6x
�2
3x2
�12
x�
13
If f
(x)
�x2
,g(x
) �
5x,a
nd
h(x
) �
x�
4,fi
nd
eac
h v
alu
e.
14.f
[g(1
)]25
15.g
[h(�
2)]
1016
.h[f
(4)]
20
17.f
[h(�
9)]
2518
.h[g
(�3)
]�
1119
.g[f
(8)]
320
20.h
[f(2
0)]
404
21.[
f�
(h�
g)](
�1)
122
.[f
�(g
�h
)](4
)16
00
23.B
USI
NES
ST
he
fun
ctio
n f
(x)
�10
00 �
0.01
x2m
odel
s th
e m
anu
fact
uri
ng
cost
per
ite
mw
hen
xit
ems
are
prod
uce
d,an
d g(
x) �
150
�0.
001x
2m
odel
s th
e se
rvic
e co
st p
er i
tem
.W
rite
a f
un
ctio
n C
(x)
for
the
tota
l m
anu
fact
uri
ng
and
serv
ice
cost
per
ite
m.
C(x
) �
1150
�0.
011x
2
24.M
EASU
REM
ENT
Th
e fo
rmu
la f
�� 1n 2�
con
vert
s in
ches
nto
fee
t f,
and
m�
� 52f 80�
con
vert
s fe
et t
o m
iles
m.W
rite
a c
ompo
siti
on o
f fu
nct
ion
s th
at c
onve
rts
inch
es t
o m
iles
.
[m�
f]n
�� 63
,n 360
�
1 � x2
Pra
ctic
e (
Ave
rag
e)
Op
erat
ion
s o
n F
un
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-7
7-7
© Glencoe/McGraw-Hill A22 Glencoe Algebra 2
Answers (Lesson 7-7)
Readin
g t
o L
earn
Math
em
ati
csO
per
atio
ns
on
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-7
7-7
©G
lenc
oe/M
cGra
w-H
ill41
5G
lenc
oe A
lgeb
ra 2
Lesson 7-7
Pre-
Act
ivit
yW
hy
is i
t im
port
ant
to c
ombi
ne
fun
ctio
ns
in b
usi
nes
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-7
at
the
top
of p
age
383
in y
our
text
book
.
Des
crib
e tw
o w
ays
to c
alcu
late
Ms.
Cof
fmon
’s p
rofi
t fr
om t
he
sale
of
50 b
irdh
ouse
s.(D
o n
ot a
ctu
ally
cal
cula
te h
er p
rofi
t.)
Sam
ple
an
swer
:1.
Fin
d t
he
reve
nu
e by
su
bst
itu
tin
g 5
0 fo
r x
in t
he
exp
ress
ion
125x
.Nex
t,fi
nd
th
e co
st b
y su
bst
itu
tin
g 5
0 fo
r x
in t
he
exp
ress
ion
65x
�54
00.F
inal
ly,s
ub
trac
t th
e co
st f
rom
th
ere
ven
ue
to f
ind
th
e p
rofi
t.2.
Fo
rm t
he
pro
fit
fun
ctio
n
p(x
) �
r(x
) �
c(x
) �
125x
�(6
5x�
5400
) �
60x
�54
00.
Su
bst
itu
te 5
0 fo
r x
in t
he
exp
ress
ion
60x
�54
00.
Rea
din
g t
he
Less
on
1.D
eter
min
e w
het
her
eac
h s
tate
men
t is
tru
eor
fal
se.(
Rem
embe
r th
at t
rue
mea
ns
alw
ays
tru
e.)
a.If
fan
d g
are
poly
nom
ial
fun
ctio
ns,
then
f�
gis
a p
olyn
omia
l fu
nct
ion
.tr
ue
b.
If f
and
gar
e po
lyn
omia
l fu
nct
ion
s,th
en
is a
pol
ynom
ial
fun
ctio
n.
fals
e
c.If
fan
d g
are
poly
nom
ial
fun
ctio
ns,
the
dom
ain
of
the
fun
ctio
n f
�g
is t
he
set
of a
llre
al n
um
bers
.tr
ue
d.
If f
(x)
�3x
�2
and
g(x)
�x
�4,
the
dom
ain
of
the
fun
ctio
n
is t
he
set
of a
ll r
eal
nu
mbe
rs.
fals
e
e.If
fan
d g
are
poly
nom
ial
fun
ctio
ns,
then
(f
�g)
(x)
�(g
�f)
(x).
fals
e
f.If
fan
d g
are
poly
nom
ial
fun
ctio
ns,
then
(f
�g)
(x)
�(g
� f)
(x)
tru
e
2.L
et f
(x)
�2x
�5
and
g(x)
�x2
�1.
a.E
xpla
in i
n w
ords
how
you
wou
ld f
ind
(f�
g)(�
3).(
Do
not
act
ual
ly d
o an
y ca
lcu
lati
ons.
)S
amp
le a
nsw
er:
Sq
uar
e �
3 an
d a
dd
1.T
ake
the
nu
mb
er y
ou
get
,m
ult
iply
it b
y 2,
and
su
btr
act
5.
b.
Exp
lain
in
wor
ds h
ow y
ou w
ould
fin
d (g
�f)
(�3)
.(D
o n
ot a
ctu
ally
do
any
calc
ula
tion
s.)
Sam
ple
an
swer
:M
ult
iply
�3
by 2
an
d s
ub
trac
t 5.
Take
th
en
um
ber
yo
u g
et,s
qu
are
it,a
nd
ad
d 1
.
Hel
pin
g Y
ou
Rem
emb
er
3.S
ome
stu
den
ts h
ave
trou
ble
rem
embe
rin
g th
e co
rrec
t or
der
in w
hic
h t
o ap
ply
the
two
orig
inal
fu
nct
ion
s w
hen
eva
luat
ing
a co
mpo
site
fu
nct
ion
.Wri
te t
hre
e se
nte
nce
s,ea
ch o
fw
hic
h e
xpla
ins
how
to
do t
his
in
a s
ligh
tly
diff
eren
t w
ay.(
Hin
t:U
se t
he
wor
d cl
oses
tin
the
firs
t se
nte
nce
,th
e w
ords
in
sid
ean
d ou
tsid
ein
th
e se
con
d,an
d th
e w
ords
lef
tan
dri
ght
in t
he
thir
d.)
Sam
ple
an
swer
:1.
Th
e fu
nct
ion
th
at is
wri
tten
clo
sest
to
the
vari
able
is a
pp
lied
fir
st.2
.Wo
rk f
rom
th
e in
sid
e to
th
e o
uts
ide.
3.W
ork
fro
m r
igh
t to
left
.
f � g
f � g
©G
lenc
oe/M
cGra
w-H
ill41
6G
lenc
oe A
lgeb
ra 2
Rel
ativ
e M
axim
um
Val
ues
Th
e gr
aph
of
f(x)
�x3
�6x
�9
show
s a
rela
tive
max
imu
m v
alu
e so
mew
her
e be
twee
n f
(�2)
an
d f(
�1)
.You
can
obt
ain
a
clos
er a
ppro
xim
atio
n b
y co
mpa
rin
g va
lues
su
ch a
s th
ose
show
n i
n t
he
tabl
e.
To
the
nea
rest
ten
th a
rel
ativ
e m
axim
um
va
lue
for
f(x)
is
�3.
3.
Usi
ng
a ca
lcu
lato
r to
fin
d p
oin
ts,g
rap
h e
ach
fu
nct
ion
.To
the
nea
rest
te
nth
,fin
d a
rel
ativ
e m
axim
um
val
ue
of t
he
fun
ctio
n.
1.f(
x) �
x(x2
�3)
rel.
max
.of
2.0
2.f(
x) �
x3�
3x�
3re
l.m
ax.o
f �
1.0
3.f(
x) �
x3�
9x�
2re
l.m
ax.o
f 8.
44.
f(x)
�x3
�2x
2�
12x
�24
rel.
max
.of
3.3
5
x
f(x)
O1
2
x
f(x)
O2
x
f(x)
O
x
f(x)
O
xf(
x)
�2
�5
�1.
5�
3.37
5
�1.
4�
3.34
4
�1.
3�
3.39
7
�1
�4
x
f(x)
O2
–2–4 –8 –12
–16
–20
–44
f(x)
� x
3 �
6x
� 9
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-7
7-7
© Glencoe/McGraw-Hill A23 Glencoe Algebra 2
An
swer
s
Answers (Lesson 7-8)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Inve
rse
Fu
nct
ion
s an
d R
elat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-8
7-8
©G
lenc
oe/M
cGra
w-H
ill41
7G
lenc
oe A
lgeb
ra 2
Lesson 7-8
Fin
d In
vers
es
Inve
rse
Rel
atio
ns
Two
rela
tions
are
inve
rse
rela
tions
if a
nd o
nly
if w
hene
ver
one
rela
tion
cont
ains
the
el
emen
t (a
, b
), t
he o
ther
rel
atio
n co
ntai
ns t
he e
lem
ent
(b,
a).
Pro
per
ty o
f In
vers
e S
uppo
se f
and
f�1
are
inve
rse
func
tions
.F
un
ctio
ns
The
n f(
a) �
bif
and
only
if f
�1 (
b) �
a.
Fin
d t
he
inve
rse
of t
he
fun
ctio
n f
(x)
�x
�.T
hen
gra
ph
th
efu
nct
ion
an
d i
ts i
nve
rse.
Ste
p 1
Rep
lace
f(x
) w
ith
yin
th
e or
igin
al e
quat
ion
.
f(x)
��2 5� x
�→
y�
�2 5� x�
Ste
p 2
Inte
rch
ange
xan
d y.
x�
�2 5� y�
Ste
p 3
Sol
ve f
or y
.
x�
�2 5� y�
Inve
rse
5x�
2y�
1M
ultip
ly e
ach
side
by
5.
5x�
1 �
2yA
dd 1
to
each
sid
e.
(5x
�1)
�y
Div
ide
each
sid
e by
2.
Th
e in
vers
e of
f(x
) �
�2 5� x�
is f
�1 (
x) �
(5x
�1)
.
Fin
d t
he
inve
rse
of e
ach
fu
nct
ion
.Th
en g
rap
h t
he
fun
ctio
n a
nd
its
in
vers
e.
1.f(
x) �
x�
12.
f(x)
�2x
�3
3.f(
x) �
x�
2
f�1 (
x)
�x
�f�
1 (x
) �
x�
f�1 (
x)
�4x
�8
f–1(x
) � 4
x �
8
f(x)
� 1 – 4x
� 2 x
f (x)
O
f(x)
� 2
x �
3
f–1(x
) � 1 – 2x
� 3 – 2
x
f (x)
O
f(x)
� 2 – 3x
� 1
f–1(x
) � 3 – 2x
� 3 – 2
x
f (x)
O
3 � 21 � 2
3 � 23 � 2
1 � 42 � 3
1 � 21 � 5
1 � 2
1 � 51 � 5
1 � 51 � 5
x
f (x)
O
f(x)
� 2 – 5x
� 1 – 5
f–1(x
) � 5 – 2x
� 1 – 2
1 � 52 � 5
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill41
8G
lenc
oe A
lgeb
ra 2
Inve
rses
of
Rel
atio
ns
and
Fu
nct
ion
s
Inve
rse
Fu
nct
ion
sTw
o fu
nctio
ns f
and
gar
e in
vers
e fu
nctio
ns if
and
onl
y if
[f�
g](
x) �
xan
d [g
�f]
(x)
�x.
Det
erm
ine
wh
eth
er f
(x)
�2x
�7
and
g(x
) �
(x�
7) a
re i
nve
rse
fun
ctio
ns.
[f�
g](x
) �
f[g(
x)]
[g�
f](x
) �
g[f(
x)]
�f ��1 2� (
x�
7)�
�g(
2x�
7)
�2 ��1 2� (
x�
7)��
7�
�1 2� (2x
�7
�7)
�x
�7
�7
�x
�x
Th
e fu
nct
ion
s ar
e in
vers
es s
ince
bot
h [
f�
g](x
) �
xan
d [g
�f]
(x)
�x.
Det
erm
ine
wh
eth
er f
(x)
�4x
�an
d g
(x)
�x
�3
are
inve
rse
fun
ctio
ns.
[f�
g](x
) �
f[g(
x)]
�f ��1 4� x
�3 �
�4 ��1 4� x
�3 �
��1 3�
�x
�12
��1 3�
�x
�11
�2 3�
Sin
ce [
f�
g](x
)
x,th
e fu
nct
ion
s ar
e n
ot i
nve
rses
.
Det
erm
ine
wh
eth
er e
ach
pai
r of
fu
nct
ion
s ar
e in
vers
e fu
nct
ion
s.
1.f(
x) �
3x�
12.
f(x)
��1 4� x
�5
3.f(
x) �
�1 2� x�
10
g(x)
��1 3� x
��1 3�
yes
g(x)
�4x
�20
yes
g(x)
�2x
�� 11 0�
no
4.f(
x) �
2x�
55.
f(x)
�8x
�12
6.f(
x) �
�2x
�3
g(x)
�5x
�2
no
g(x)
��1 8� x
�12
no
g(x)
��
�1 2� x�
�3 2�ye
s
7.f(
x) �
4x�
�1 2�8.
f(x)
�2x
��3 5�
9.f(
x) �
4x�
�1 2�
g(x)
��1 4� x
��1 8�
yes
g(x)
�� 11 0�
(5x
�3)
yes
g(x)
��1 2� x
��3 2�
no
10.f
(x)
�10
�� 2x �
11.f
(x)
�4x
��4 5�
12.f
(x)
�9
��3 2� x
g(x)
�20
�2x
yes
g(x)
�� 4x �
��1 5�
yes
g(x)
��2 3� x
�6
yes
1 � 41 � 3
1 � 2
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Inve
rse
Fu
nct
ion
s an
d R
elat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-8
7-8
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A24 Glencoe Algebra 2
Answers (Lesson 7-8)
Skil
ls P
ract
ice
Inve
rse
Fu
nct
ion
s an
d R
elat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-8
7-8
©G
lenc
oe/M
cGra
w-H
ill41
9G
lenc
oe A
lgeb
ra 2
Lesson 7-8
Fin
d t
he
inve
rse
of e
ach
rel
atio
n.
1.{(
3,1)
,(4,
�3)
,(8,
�3)
}2.
{(�
7,1)
,(0,
5),(
5,�
1)}
{(1,
3),(
�3,
4),(
�3,
8)}
{(1,
�7)
,(5,
0),(
�1,
5)}
3.{(
�10
,�2)
,(�
7,6)
,(�
4,�
2),(
�4,
0)}
4.{(
0,�
9),(
5,�
3),(
6,6)
,(8,
�3)
}{(
�2,
�10
),(6
,�7)
,(�
2,�
4),(
0,�
4)}
{(�
9,0)
,(�
3,5)
,(6,
6),(
�3,
8)}
5.{(
�4,
12),
(0,7
),(9
,�1)
,(10
,�5)
}6.
{(�
4,1)
,(�
4,3)
,(0,
�8)
,(8,
�9)
}{(
12,�
4),(
7,0)
,(�
1,9)
,(�
5,10
)}{(
1,�
4),(
3,�
4),(
�8,
0),(
�9,
8)}
Fin
d t
he
inve
rse
of e
ach
fu
nct
ion
.Th
en g
rap
h t
he
fun
ctio
n a
nd
its
in
vers
e.
7.y
�4
8.f(
x) �
3x9.
f(x)
�x
�2
x�
4f�
1 (x
) �
�1 3� xf�
1 (x
) �
x�
2
10.g
(x)
�2x
�1
11.h
(x)
��1 4� x
12.y
��2 3� x
�2
g�
1 (x
) �
�x� 2
1�
h�
1 (x
) �
4xy
��3 2� x
�3
Det
erm
ine
wh
eth
er e
ach
pai
r of
fu
nct
ion
s ar
e in
vers
e fu
nct
ion
s.
13.f
(x)
�x
�1
no
14.f
(x)
�2x
�3
yes
15.f
(x)
�5x
�5
yes
g(x)
�1
�x
g(x)
��1 2� (
x�
3)g(
x) �
�1 5� x�
1
16.f
(x)
�2x
yes
17.h
(x)
�6x
�2
no
18.f
(x)
�8x
�10
yes
g(x)
��1 2� x
g(x)
��1 6� x
�3
g(x)
��1 8� x
��5 4�
x
y
Ox
h (x)
Ox
g (x)
O
x
f (x)
Ox
f (x)
Ox
y
O
©G
lenc
oe/M
cGra
w-H
ill42
0G
lenc
oe A
lgeb
ra 2
Fin
d t
he
inve
rse
of e
ach
rel
atio
n.
1.{(
0,3)
,(4,
2),(
5,�
6)}
2.{(
�5,
1),(
�5,
�1)
,(�
5,8)
}{(
3,0)
,(2,
4),(
�6,
5)}
{(1,
�5)
,(�
1,�
5),(
8,�
5)}
3.{(
�3,
�7)
,(0,
�1)
,(5,
9),(
7,13
)}4.
{(8,
�2)
,(10
,5),
(12,
6),(
14,7
)}{(
�7,
�3)
,(�
1,0)
,(9,
5),(
13,7
)}{(
�2,
8),(
5,10
),(6
,12)
,(7,
14)}
5.{(
�5,
�4)
,(1,
2),(
3,4)
,(7,
8)}
6.{(
�3,
9),(
�2,
4),(
0,0)
,(1,
1)}
{(�
4,�
5),(
2,1)
,(4,
3),(
8,7)
}{(
9,�
3),(
4,�
2),(
0,0)
,(1,
1)}
Fin
d t
he
inve
rse
of e
ach
fu
nct
ion
.Th
en g
rap
h t
he
fun
ctio
n a
nd
its
in
vers
e.
7.f(
x) �
�3 4� x8.
g(x)
�3
�x
9.y
�3x
�2
f�1 (
x)
��4 3� x
g�
1 (x
) �
x�
3y
��x
� 32
�
Det
erm
ine
wh
eth
er e
ach
pai
r of
fu
nct
ion
s ar
e in
vers
e fu
nct
ion
s.
10.f
(x)
�x
�6
yes
11.f
(x)
��
4x�
1ye
s12
.g(x
) �
13x
�13
no
g(x)
�x
�6
g(x)
��1 4� (
1 �
x)h
(x)
�� 11 3�
x�
1
13.f
(x)
�2x
no
14.f
(x)
��6 7� x
yes
15.g
(x)
�2x
�8
yes
g(x)
��
2xg(
x) �
�7 6� xh
(x)
��1 2� x
�4
16. M
EASU
REM
ENT
Th
e po
ints
(63
,121
),(7
1,18
0),(
67,1
40),
(65,
108)
,an
d (7
2,16
5) g
ive
the
wei
ght
in p
oun
ds a
s a
fun
ctio
n o
f h
eigh
t in
in
ches
for
5 s
tude
nts
in
a c
lass
.Giv
e th
epo
ints
for
th
ese
stu
den
ts t
hat
rep
rese
nt
hei
ght
as a
fu
nct
ion
of
wei
ght.
(121
,63)
,(18
0,71
),(1
40,6
7),(
108,
65),
(165
,72)
REM
OD
ELIN
GF
or E
xerc
ises
17
and
18,
use
th
e fo
llow
ing
info
rmat
ion
.T
he
Cle
arys
are
rep
laci
ng
the
floo
rin
g in
th
eir
15 f
oot
by 1
8 fo
ot k
itch
en.T
he
new
flo
orin
gco
sts
$17.
99 p
er s
quar
e ya
rd.T
he
form
ula
f(x
) �
9xco
nve
rts
squ
are
yard
s to
squ
are
feet
.
17.F
ind
the
inve
rse
f�1 (
x).W
hat
is
the
sign
ific
ance
of
f�1 (
x) f
or t
he
Cle
arys
?f�
1 (x
) �
�x 9� ;It
will
allo
w t
hem
to
co
nver
t th
e sq
uar
e fo
ota
ge
of
thei
r ki
tch
en f
loo
r to
squ
are
yard
s,so
th
ey c
an t
hen
cal
cula
te t
he
cost
of
the
new
flo
ori
ng
.
18.W
hat
wil
l th
e n
ew f
loor
ing
cost
th
e C
lear
y’s?
$539
.70
x
y
Ox
g (x)
Ox
f (x)
OPra
ctic
e (
Ave
rag
e)
Inve
rse
Fu
nct
ion
s an
d R
elat
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-8
7-8
© Glencoe/McGraw-Hill A25 Glencoe Algebra 2
An
swer
s
Answers (Lesson 7-8)
Readin
g t
o L
earn
Math
em
ati
csIn
vers
e F
un
ctio
ns
and
Rel
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-8
7-8
©G
lenc
oe/M
cGra
w-H
ill42
1G
lenc
oe A
lgeb
ra 2
Lesson 7-8
Pre-
Act
ivit
yH
ow a
re i
nve
rse
fun
ctio
ns
rela
ted
to
mea
sure
men
t co
nve
rsio
ns?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-8
at
the
top
of p
age
390
in y
our
text
book
.
A f
unct
ion
mul
tipl
ies
a nu
mbe
r by
3 a
nd t
hen
adds
5 t
o th
e re
sult
.Wha
t do
esth
e in
vers
e fu
nct
ion
do,
and
in w
hat
ord
er?
Sam
ple
an
swer
:It
fir
stsu
btr
acts
5 f
rom
th
e n
um
ber
an
d t
hen
div
ides
th
e re
sult
by
3.
Rea
din
g t
he
Less
on
1.C
ompl
ete
each
sta
tem
ent.
a.If
tw
o re
lati
ons
are
inve
rses
,th
e do
mai
n o
f on
e re
lati
on i
s th
e of
the
oth
er.
b.
Su
ppos
e th
at g
(x)
is a
rel
atio
n a
nd
that
th
e po
int
(4,�
2) i
s on
its
gra
ph.T
hen
a p
oin
t
on t
he
grap
h o
f g�
1 (x)
is
.
c.T
he
test
can
be
use
d on
th
e gr
aph
of
a fu
nct
ion
to
dete
rmin
e
wh
eth
er t
he
fun
ctio
n h
as a
n i
nve
rse
fun
ctio
n.
d.
If y
ou a
re g
iven
th
e gr
aph
of
a fu
nct
ion
,you
can
fin
d th
e gr
aph
of
its
inve
rse
by
refl
ecti
ng
the
orig
inal
gra
ph o
ver
the
lin
e w
ith
equ
atio
n
.
e.If
fan
d g
are
inve
rse
fun
ctio
ns,
then
(f
�g)
(x)
�
and
(g�
f)(x
) �
.
f.A
fu
nct
ion
has
an
in
vers
e th
at i
s al
so a
fu
nct
ion
on
ly i
f th
e gi
ven
fu
nct
ion
is
.
g.S
upp
ose
that
h(x
) is
a f
un
ctio
n w
hos
e in
vers
e is
als
o a
fun
ctio
n.I
f h
(5)
�12
,th
enh
�1 (
12)
�.
2.A
ssu
me
that
f(x
) is
a o
ne-
to-o
ne
fun
ctio
n d
efin
ed b
y an
alg
ebra
ic e
quat
ion
.Wri
te t
he
fou
rst
eps
you
wou
ld f
ollo
w i
n o
rder
to
fin
d th
e eq
uat
ion
for
f�
1 (x)
.
1.R
epla
ce f
(x)
wit
h y
in t
he
ori
gin
al e
qu
atio
n.
2.In
terc
han
ge
xan
d y
.
3.S
olv
e fo
r y.
4.R
epla
ce y
wit
h f
�1 (
x).
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
som
eth
ing
new
is
to r
elat
e it
to
som
eth
ing
you
alr
eady
kn
ow.
How
are
the
ver
tica
l an
d ho
rizo
ntal
lin
e te
sts
rela
ted?
Sam
ple
an
swer
:Th
e ve
rtic
allin
e te
st d
eter
min
es w
het
her
a r
elat
ion
is a
fu
nct
ion
bec
ause
th
e o
rder
edp
airs
in a
fu
nct
ion
can
hav
e n
o r
epea
ted
x-v
alu
es.T
he
ho
rizo
nta
l lin
e te
std
eter
min
es w
het
her
a f
un
ctio
n is
on
e-to
-on
e b
ecau
se a
on
e-to
-on
efu
nct
ion
can
no
t h
ave
any
rep
eate
d y
-val
ues
.
5
on
e-to
-on
e
xx
y �
x
ho
rizo
nta
l lin
e
(�2,
4)
ran
ge
©G
lenc
oe/M
cGra
w-H
ill42
2G
lenc
oe A
lgeb
ra 2
Min
iatu
re G
olf
In m
inia
ture
gol
f,th
e ob
ject
of
the
gam
e is
to
roll
th
e go
lf b
all
into
th
e h
ole
in a
s fe
w s
hot
s as
pos
sibl
e.A
s in
th
e di
agra
m a
t th
e ri
ght,
the
hol
e is
oft
en p
lace
d so
th
at a
dir
ect
shot
is
impo
ssib
le.R
efle
ctio
ns
can
be
use
d to
hel
p de
term
ine
the
dire
ctio
n t
hat
th
e ba
ll s
hou
ld b
ero
lled
in
ord
er t
o sc
ore
a h
ole-
in-o
ne.
Usi
ng
wal
l E �
F�,f
ind
th
e p
ath
to
use
to
sc
ore
a h
ole-
in-o
ne.
Fin
d th
e re
flec
tion
im
age
of t
he
“hol
e”w
ith
res
pect
to
E�F�
and
labe
l it
H .
Th
e in
ters
ecti
on o
f B �
H� �w
ith
wal
l E�
F�is
th
e po
int
at w
hic
h t
he
shot
sh
ould
be
dire
cted
.
For
th
e h
ole
at t
he
righ
t,fi
nd
a p
ath
to
scor
e a
hol
e-in
-on
e.
Fin
d th
e re
flec
tion
im
age
of H
wit
h r
espe
ct t
o E�
F�an
d la
bel
it H
.In
th
is c
ase,
B �H�
�in
ters
ects
J�K�
befo
re i
nte
rsec
tin
g E�
F�.T
hu
s,th
ispa
th c
ann
ot b
e u
sed.
To
fin
d a
usa
ble
path
,fin
d th
e re
flec
tion
im
age
of H
w
ith
res
pect
to
G �F�
and
labe
l it
H�.
Now
,th
e in
ters
ecti
on o
f B �
H���
wit
h w
all
G�F�
is t
he
poin
t at
wh
ich
th
e sh
otsh
ould
be
dire
cted
.
Cop
y ea
ch f
igu
re.T
hen
,use
ref
lect
ion
s to
det
erm
ine
a p
ossi
ble
p
ath
for
a h
ole-
in-o
ne.
1.2.
3.
H
B
H
B
H
B
B GF
JK
H' H"
E
H
Bal
l
Hol
e
E
H'
F
Bal
l
Hol
e
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-8
7-8
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
© Glencoe/McGraw-Hill A26 Glencoe Algebra 2
Answers (Lesson 7-9)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Sq
uar
e R
oo
t F
un
ctio
ns
and
Ineq
ual
itie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-9
7-9
©G
lenc
oe/M
cGra
w-H
ill42
3G
lenc
oe A
lgeb
ra 2
Lesson 7-9
Squ
are
Ro
ot
Fun
ctio
ns
A f
un
ctio
n t
hat
con
tain
s th
e sq
uar
e ro
ot o
f a
vari
able
expr
essi
on i
s a
squ
are
root
fu
nct
ion
.
Gra
ph
y�
�3x
��
2�.S
tate
its
dom
ain
an
d r
ange
.
Sin
ce t
he
radi
can
d ca
nn
ot b
e n
egat
ive,
3x�
2 �
0 or
x�
�2 3� .
Th
e x-
inte
rcep
t is
�2 3� .T
he
ran
ge i
s y
�0.
Mak
e a
tabl
e of
val
ues
an
d gr
aph
th
e fu
nct
ion
.
Gra
ph
eac
h f
un
ctio
n.S
tate
th
e d
omai
n a
nd
ran
ge o
f th
e fu
nct
ion
.
1.y
��
2x�2.
y�
�3�
x�3.
y�
���� 2x �
D:
x�
0;R
:y
�0
D:
x�
0;R
:y
�0
D:
x�
0;R
:y
�0
4.y
�2�
x�
3�
5.y
��
�2x
�3
�6.
y�
�2x
�5
�
D:
x�
3;R
:y
�0
D:
x�
�3 2� ;R
:y
�0
D:
x�
��5 2� ;
R:
y�
0
y �
��
�2x
� 5
x
y
O
y �
��
��
�2x
� 3
x
y
O
y �
2�
��
x �
3
x
y
O
y �
���x – 2
x
y
Oy
� �
3��x
xy
O
y �
��2x
x
y
O
xy
�2 3�0
11
22
3�
7�
x
y
O
y �
��
��
3x �
2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill42
4G
lenc
oe A
lgeb
ra 2
Squ
are
Ro
ot
Ineq
ual
itie
sA
sq
uar
e ro
ot i
neq
ual
ity
is a
n i
neq
ual
ity
that
con
tain
sth
e sq
uar
e ro
ot o
f a
vari
able
exp
ress
ion
.Use
wh
at y
ou k
now
abo
ut
grap
hin
g sq
uar
e ro
otfu
nct
ion
s an
d qu
adra
tic
ineq
ual
itie
s to
gra
ph s
quar
e ro
ot i
neq
ual
itie
s.
Gra
ph
y�
�2x
��
1��
2.G
raph
th
e re
late
d eq
uat
ion
y�
�2x
�1
��
2.S
ince
th
e bo
un
dary
sh
ould
be
incl
ude
d,th
e gr
aph
sh
ould
be
soli
d.
Th
e do
mai
n i
ncl
ude
s va
lues
for
x�
�1 2� ,so
th
e gr
aph
is
to t
he
righ
t
of x
��1 2� .
Th
e ra
nge
in
clu
des
only
nu
mbe
rs g
reat
er t
han
2,s
o th
e
grap
h i
s ab
ove
y�
2.
Gra
ph
eac
h i
neq
ual
ity.
1.y
�2�
x�2.
y�
�x
�3
�3.
y�
3�2x
�1
�
4.y
��
3x�
4�
5.y
��
x�
1�
�4
6.y
�2�
2x�
3�
7.y
��
3x�
1�
�2
8.y
��
4x�
2�
�1
9.y
�2�
2x�
1�
�4
y �
2�
���
2x �
1 �
4 x
y
O
y �
���
�4x
� 2
� 1 x
y
Oy
� �
��
�3x
� 1
� 2
x
y
O
y �
2�
��
�2x
� 3 x
y
Oy
� �
��
x �
1 �
4
x
y
O
y �
���
�3x
� 4
x
y
O
y �
3�
���
2x �
1
x
y
O
y �
��
�x
� 3
x
y
O
y �
2�
�x
x
y
O
x
y
O
y �
��
��
2x �
1 �
2
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Sq
uar
e R
oo
t F
un
ctio
ns
and
Ineq
ual
itie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-9
7-9
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A27 Glencoe Algebra 2
An
swer
s
Answers (Lesson 7-9)
Skil
ls P
ract
ice
Sq
uar
e R
oo
t F
un
ctio
ns
and
Ineq
ual
itie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-9
7-9
©G
lenc
oe/M
cGra
w-H
ill42
5G
lenc
oe A
lgeb
ra 2
Lesson 7-9
Gra
ph
eac
h f
un
ctio
n.S
tate
th
e d
omai
n a
nd
ran
ge o
f ea
ch f
un
ctio
n.
1.y
��
2x�2.
y�
��
3x�3.
y�
2�x�
D:
x�
0,R
:y
�0
D:
x�
0,R
:y
�0
D:
x�
0,R
:y
�0
4.y
��
x�
3�
5.y
��
�2x
�5
�6.
y�
�x
�4
��
2
D:
x�
�3,
R:
y�
0D
:x
�2.
5,R
:y
�0
D:
x�
�4,
R:
y�
�2
Gra
ph
eac
h i
neq
ual
ity.
7.y
��
4x�8.
y�
�x
�1
�9.
y�
�4x
�3
�
x
y
Ox
y
Ox
y
O
x
y
Ox
y
O
x
y
O
x
y
O
x
y
Ox
y
O
©G
lenc
oe/M
cGra
w-H
ill42
6G
lenc
oe A
lgeb
ra 2
Gra
ph
eac
h f
un
ctio
n.S
tate
th
e d
omai
n a
nd
ran
ge o
f ea
ch f
un
ctio
n.
1.y
��
5x�2.
y�
��
x�
1�
3.y
�2�
x�
2�
D:
x�
0,R
:y
�0
D:
x�
1,R
:y
�0
D:
x�
�2,
R:
y�
0
4.y
��
3x�
4�
5.y
��
x�
7�
�4
6.y
�1
��
2x�
3�
D:
x�
�4 3� ,R
:y
�0
D:
x�
�7,
R:
y�
�4
D:
x�
��3 2� ,
R:
y�
1
Gra
ph
eac
h i
neq
ual
ity.
7.y
��
�6x�
8.y
��
x�
5�
�3
9.y
��
2�3x
�2
�
10.R
OLL
ER C
OA
STER
ST
he
velo
city
of
a ro
ller
coa
ster
as
it m
oves
dow
n a
hil
l is
v
��
v 02
��
64h
�,w
her
e v 0
is t
he
init
ial
velo
city
an
d h
is t
he
vert
ical
dro
p in
fee
t.If
v
�70
fee
t pe
r se
con
d an
d v 0
�8
feet
per
sec
ond,
fin
d h
.ab
ou
t 75
.6 f
t
11.W
EIG
HT
Use
th
e fo
rmu
la d
���
�39
60,w
hic
h r
elat
es d
ista
nce
fro
m E
arth
d
in m
iles
to
wei
ght.
If a
n a
stro
nau
t’s w
eigh
t on
Ear
th W
Eis
148
pou
nds
an
d in
spa
ce W
sis
115
pou
nds
,how
far
fro
m E
arth
is
the
astr
onau
t?ab
ou
t 53
2 m
i
3960
2W
E�
� Ws
x
y O
x
y
O
x
y
O
x
y Ox
y
O
x
y
O
x
y
O
x
y
O
x
y
O
Pra
ctic
e (
Ave
rag
e)
Sq
uar
e R
oo
t F
un
ctio
ns
and
Ineq
ual
itie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-9
7-9
© Glencoe/McGraw-Hill A28 Glencoe Algebra 2
Answers (Lesson 7-9)
Readin
g t
o L
earn
Math
em
ati
csS
qu
are
Ro
ot
Fu
nct
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-9
7-9
©G
lenc
oe/M
cGra
w-H
ill42
7G
lenc
oe A
lgeb
ra 2
Lesson 7-9
Pre-
Act
ivit
yH
ow a
re s
qu
are
root
fu
nct
ion
s u
sed
in
bri
dge
des
ign
?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-9
at
the
top
of p
age
395
in y
our
text
book
.
If t
he
wei
ght
to b
e su
ppor
ted
by a
ste
el c
able
is
dou
bled
,sh
ould
th
edi
amet
er o
f th
e su
ppor
t ca
ble
also
be
dou
bled
? If
not
,by
wh
at n
um
ber
shou
ld t
he
diam
eter
be
mu
ltip
lied
?
no
;�
2�
Rea
din
g t
he
Less
on
1.M
atch
eac
h s
quar
e ro
ot f
un
ctio
n f
rom
th
e li
st o
n t
he
left
wit
h i
ts d
omai
n a
nd
ran
ge f
rom
the
list
on
th
e ri
ght.
a.y
��
x�iv
i.do
mai
n:x
�0;
ran
ge:y
�3
b.
y�
�x
�3
�vi
iiii
.do
mai
n:x
�0;
ran
ge:y
�0
c.y
��
x��
3i
iii.
dom
ain
:x�
0;ra
nge
:y�
�3
d.
y�
�x
�3
�v
iv.
dom
ain
:x�
0;ra
nge
:y�
0
e.y
��
�x�
iiv.
dom
ain
:x�
3;ra
nge
:y�
0
f.y
��
�x
�3
�vi
ivi
.do
mai
n:x
�3;
ran
ge:y
�3
g.y
��
3 �
x�
�3
vivi
i.do
mai
n:x
�3;
ran
ge:y
�0
h.
y�
��
x��
3iii
viii
.do
mai
n:x
��
3;ra
nge
:y�
0
2.T
he
grap
h o
f th
e in
equ
alit
y y
��
3x�
6�
is a
sh
aded
reg
ion
.Wh
ich
of
the
foll
owin
gpo
ints
lie
in
side
th
is r
egio
n?
(3,0
)(2
,4)
(5,2
)(4
,�2)
(6,6
)
(3,0
),(5
,2),
(4,�
2)
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
som
eth
ing
is t
o ex
plai
n i
t to
som
eon
e el
se.S
upp
ose
you
are
stu
dyin
g th
is l
esso
n w
ith
a c
lass
mat
e w
ho
thin
ks t
hat
you
can
not
hav
e sq
uar
e ro
otfu
nct
ion
s be
cau
se e
very
pos
itiv
e re
al n
um
ber
has
tw
o sq
uar
e ro
ots.
How
wou
ld y
ouex
plai
n t
he
idea
of
squ
are
root
fu
nct
ion
s to
you
r cl
assm
ate?
Sam
ple
an
swer
:To
fo
rm a
sq
uar
e ro
ot
fun
ctio
n,c
ho
ose
eit
her
the
po
siti
ve o
r n
egat
ive
squ
are
roo
t.F
or
exam
ple
,y�
�x�
and
y�
��
x�ar
etw
o s
epar
ate
fun
ctio
ns.
©G
lenc
oe/M
cGra
w-H
ill42
8G
lenc
oe A
lgeb
ra 2
Rea
din
g A
lgeb
raIf
tw
o m
ath
emat
ical
pro
blem
s h
ave
basi
c st
ruct
ura
l si
mil
arit
ies,
they
are
sai
d to
be
anal
ogou
s.U
sin
g an
alog
ies
is o
ne
way
of
disc
over
ing
and
prov
ing
new
th
eore
ms.
Th
e fo
llow
ing
nu
mbe
red
sen
ten
ces
disc
uss
a t
hre
e-di
men
sion
alan
alog
y to
th
e P
yth
agor
ean
th
eore
m.
01C
onsi
der
a te
trah
edro
n w
ith
th
ree
perp
endi
cula
r fa
ces
that
mee
t at
ver
tex
O.
02S
uppo
se y
ou w
ant
to k
now
how
the
are
as A
,B,a
nd C
ofth
e th
ree
face
s th
at m
eet
at v
erte
x O
are
rela
ted
to t
he
area
Dof
th
e fa
ce o
ppos
ite
vert
ex O
.03
It i
s n
atu
ral
to e
xpec
t a
form
ula
an
alog
ous
to t
he
Pyt
hag
orea
n t
heo
rem
z2
�x2
�y2
,wh
ich
is
tru
e fo
r a
sim
ilar
sit
uat
ion
in
tw
o di
men
sion
s.04
To
expl
ore
the
thre
e-di
men
sion
al c
ase,
you
mig
ht
gues
s a
form
ula
an
d th
en t
ry t
o pr
ove
it.
05T
wo
reas
onab
le g
ues
ses
are
D3
�A
3�
B3
�C
3an
d D
2�
A2
�B
2�
C2 .
Ref
er t
o th
e n
um
ber
ed s
ente
nce
s to
an
swer
th
e q
ues
tion
s.
1.U
se s
ente
nce
01
and
the
top
diag
ram
.Th
e pr
efix
tet
ra-
mea
ns
fou
r.W
rite
an
info
rmal
def
init
ion
of
tetr
ahed
ron
.
a th
ree-
dim
ensi
on
al f
igu
re w
ith
fo
ur
face
s
2.U
se s
ente
nce
02
and
the
top
diag
ram
.Wh
at a
re t
he
len
gth
s of
th
e si
des
ofea
ch f
ace
of t
he
tetr
ahed
ron
?a,
b,a
nd
c;
a,q
,an
d r
;b
,p,a
nd
r;
c,p
,an
d q
3.R
ewri
te s
ente
nce
01
to s
tate
a t
wo-
dim
ensi
onal
an
alog
ue.
Co
nsi
der
a t
rian
gle
wit
h t
wo
per
pen
dic
ula
r si
des
th
at m
eet
at v
erte
x C
.
4.R
efer
to
the
top
diag
ram
an
d w
rite
exp
ress
ion
s fo
r th
e ar
eas
A,B
,an
d C
men
tion
ed i
n s
ente
nce
02.
Po
ssib
le a
nsw
er:
A�
�1 2� pr,
B�
�1 2� pq
,C�
�1 2� rq
5.T
o ex
plor
e th
e th
ree-
dim
ensi
onal
cas
e,yo
u m
igh
t be
gin
by
expr
essi
ng
a,b,
and
cin
ter
ms
of p
,q,a
nd
r.U
se t
he
Pyt
hag
orea
n t
heo
rem
to
do t
his
.
a2
�q
2�
r2,b
2�
r2�
p2,c
2�
p2
�q
2
6.W
hic
h g
ues
s in
sen
ten
ce 0
5 se
ems
mor
e li
kely
? Ju
stif
y yo
ur
answ
er.
See
stu
den
ts’e
xpla
nat
ion
s.
y O
z
x
b
c
Op
a
qr
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-9
7-9
© Glencoe/McGraw-Hill A29 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. D
A
D
B
C
B
A
C
C
D
B
4x � 3
A
D
A
D
D
B
C
A
A
D
A
D
B
C
A
A
B
C
A
D
Chapter 7 Assessment Answer Key Form 1 Form 2APage 429 Page 430 Page 431
An
swer
s
(continued on the next page)
© Glencoe/McGraw-Hill A30 Glencoe Algebra 2
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: 3
D
A
B
B
D
C
A
A
D
B
A
C
B
C
C
B
A
D
A
C
�2
B
B
C
C
A
D
B
C
B
Chapter 7 Assessment Answer Key Form 2A (continued) Form 2BPage 432 Page 433 Page 434
© Glencoe/McGraw-Hill A31 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: 2
y
xO
y
xO
yes
f �1(x) � �15
�x � 2
2x2 � 1
2
�x 3 � 7x2 � 4x � 28
�2, 1, �32
�
�1, �2, �4, �8, ��12
�
5, �2i, 2i
3 or 1; 1; 2 or 0
x � 3, x � 5
�3
�15�, ��15�,i �3�, �i �3�
9(n3)2 � 36(n3)
Sample answer: rel. max. at x � �1,
rel. min. at x � 1
xO
f(x )
between �2 and �1,between 0 and 1,between 1 and 2
even; 4
x2 � x � 3
�176
Chapter 7 Assessment Answer Key Form 2CPage 435 Page 436
An
swer
s
© Glencoe/McGraw-Hill A32 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: �4
y
xO
y
xO
D: x � �2, R: y � 0
yes
g�1(x) � ��12
�x � 2
x2 � 6x � 5
4
�x3 � 6x2 � 4x � 24
�3, �23
�, 1
�1, �2, �7, �14, ��12
�, ��72
�
2, �3i, 3i
2 or 0; 2 or 0; 4, 2, or 0
x � 3, x � 1
132
��6�, �6�, i �2�, �i �2�
5(x5)2 � 4(x5) � 3
Sample answer: rel. max. at x � 1, rel. min. at x � �1
xO
f(x )
between �2 and �1,between �1 and 0,between 1 and 2
even; 4
x2 � 2x � 1
�134
Chapter 7 Assessment Answer Key Form 2DPage 437 Page 438
© Glencoe/McGraw-Hill A33 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B: 24
y
x
O
y
xO
D: x � �4, R: y � �2
yes
h�1(x) � �5x
2� 6�
960
27x3 � 9x2 � 3x � 1
x2 � 2x
��14
�, ��12
�, �13
�, 2
�1, �3, �5, �15,
��19
�, ��13
�, ��59
�, ��53
�
�1, 3, 3 � 2i, 3 � 2i
5, 3, or 1; 5, 3, or 1; 10, 8, 6, 4, 2, or 0
�3
7014
16, 81
b[9(b2)2 � 3(b2) � 8]
Sample answer: rel.max. at x � �1 and
x � 1, rel. min. at x � 0
xO
f(x )
between 0 and 1,between 1 and 2
f (x ) → �� as x → ��,
f (x) → �� as x → ��; odd; 4
x4 � 7x2 � x
�133�
An
swer
s
Chapter 7 Assessment Answer Key Form 3Page 439 Page 440
An
swer
s
© Glencoe/McGraw-Hill A34 Glencoe Algebra 2
Chapter 7 Assessment Answer KeyPage 441, Open-Ended Assessment
Scoring Rubric
Score General Description Specific Criteria
• Shows thorough understanding of the concepts ofpolynomial functions; graphing polynomial functions;determining number and type of roots of a polynomialequation; finding rational zeros of a polynomial function;operations with functions; and finding inverse of a function.
• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Graphs are accurate and appropriate.• Goes beyond requirements of some or all problems.
• Shows an understanding of the concepts of polynomialfunctions; graphing polynomial functions; determiningnumber and type of roots of a polynomial equation; findingrational zeros of a polynomial function; operations withfunctions; and finding inverse of a function.
• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.
• Shows an understanding of most of the concepts ofpolynomial functions; graphing polynomial functions;determining number and type of roots of a polynomialequation; finding rational zeros of a polynomial function;operations with functions; and finding inverse of a function.
• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Graphs are mostly accurate.• Satisfies the requirements of most of the problems.
• Final computation is correct.• No written explanations or work is shown to substantiate
the final computation.• Graphs may be accurate but lack detail or explanation.• Satisfies minimal requirements of some of the problems.
• Shows little or no understanding of most of the concepts ofpolynomial functions; graphing polynomial functions;determining number and type of roots of a polynomialequation; finding rational zeros of a polynomial function;operations with functions; and finding inverse of a function.
• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Graphs are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer may be given.
0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given
1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation
2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem
3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation
4 SuperiorA correct solution that is supported by well-developed, accurateexplanations
Chapter 7 Assessment Answer Key Page 441, Open-Ended Assessment
Sample Answers
© Glencoe/McGraw-Hill A35 Glencoe Algebra 2
1a. and b. Students must sketch a polynomial function having exactly 5 zeros, opposite end behavior,2 relative maxima, and 2 relative minima, labeled as shown.
1c. Regardless of the function sketched,D: all real numbers and R: all real numbers.
1d. Students must state either “As x → ��, f(x)→ �� and as x → ��, f(x) → ��” (as for the samplefunction shown) or “As x → ��, f(x) → �� and as x → ��,f(x) → ��.”
2. Since the range is g(x) � 2, the graph ofg(x) lies entirely on or above thehorizontal line g(x) � 2. Students shouldindicate that this would require that thedegree of g(x) must be even, that theleading coefficient must be positive, andthat the function has no zeros.
3a. Answers must be of the formP(x) � a0x4 � a1x3 � a2x2 � a3x � a4,where an � 0 for any n. Sampleanswer: P(x) � x4 � x3 � x2 � 2x � 3.
3b. Students should show by directsubstitution, and by syntheticsubstitution, how to find P(�2).For the sample function in a,P(�2) � 11.
3c. Students should indicate that x � 1 isa factor of P(x) if and only if P(�1) � 0.For the sample function in a,P(�1) � 2 � 0, so x � 1 is not a factorof P(x).
3d. Students should use Descartes’ Rule ofSigns to determine the number ofpositive and negative real zeros ofP(x). For the sample function in a,P(x) has no positive real zeros and has4, 2 or 0 negative real zeros.
3e. Students must explain that anyrational zeros of P(x) must be of the form �
pq�, where p is a factor of
a4 and q is a factor of a0.For the sample function in a, a0 � 1and a4 � 3, so the only possiblerational zeros are �1 and �3.
3f. For each of the possible rational zeroszn found in part e, students must showwhether x � zn is a factor of P(x). Forthe sample function in a, there are norational zeros.
4a. Sample answers: For g(x) � x � 1 andh(x) � x2, the answers would be:2x � 4; 9a2; x2 � x � 1; 1 � x � x2;
x3 � x2; �xx�
2
1� for x � �1; x2 � 2x � 1;
x2 � 1; 16; x � 1.4b. Students should show that, for their
functions g(x) and g�1(x),g�1 � g(x) � x. Students should thenindicate that the graphs of inversefunctions are reflections of one anotherover the line y � x.
xO
A1
z1 z2 z3
z4 z5
A2
B1
B2
f(x )
In addition to the scoring rubric found on page A34, the following sample answers may be used as guidance in evaluating open-ended assessment items.
An
swer
s
© Glencoe/McGraw-Hill A36 Glencoe Algebra 2
Chapter 7 Assessment Answer Key Vocabulary Test/Review Quiz (Lessons 7–1 through 7–3) Quiz (Lessons 7–6 and 7–7)
Page 442 Page 443 Page 444
1. syntheticsubstitution
2. one-to-one function
3. quadratic form
4. fundamentaltheorem of algebra
5. square root
6. location principle
7. minimum
8. depressedpolynomial
9. composition offunctions
10. polynomial in onevariable
11. Sample answer: Theend behavior of agraph is adescription of howthe graph behaveswhen the value of xbecomes very smallor very large.
12. Sample answer: TheFactor Theorem isthe theorem thatstates that thebinomial x � a is afactor of thepolynomial f(x) ifand only if f(a) � 0.
1.
2.
3.
4.
5.
Quiz (Lessons 7–4 and 7–5)
Page 443
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
Quiz (Lessons 7–8 and 7–9)
Page 444
1.
2.
3.
4.y
xO
D: x � �3, R: y � 0
yes
x
f(x)
f –1(x)
f (x)
O
f�1(x) � �x �
42
�
{(5, �2), (4, 0), (�8, 1), (7, 4)}
28; 122
x2 � 6x � 7; x2 � 2x � 5
{(2, 4), (3, 8), (4, 3), (8, 4)}; {(2, 5), (4, 2), (5, 4)}
x2 � x � 6; x2 � 5x � 2;2x3 � 2x2 � 8x � 8; �x2
2�x
3�x
4� 2
�, x �2
�1, �2, �3, �6,
��12
�, ��32
�; �1, �32
�
B
3, 1 � i, 1 � i
3 or 1; 2 or 0; 4, 2, or 0
x � 2, x � 3
46, 277
��7�, �7�,�i �7�, i �7�
�3, ��5�, �5�, 3
xO
f(x )
between �1 and 0,between 1 and 2,between 3 and 4
even; 2
57
© Glencoe/McGraw-Hill A37 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.f�1(x) � �
x �5
4�
�3, �12
�, 2
278
x2 � x � 6 � 0
x � 13
(2n � 5)2
(�3, 5)
�1, �4i, 4i
f(x) � x3 � 3x2 � 9x � 27
2 or 0; 2 or 0; 4, 2, or 0
�3, 0, 3, �3i, 3i
xO
f(x )
at x � 1, between �4 and�3, between �2 and �1
D
A
C
D
A
B
Chapter 7 Assessment Answer Key Mid-Chapter Test Cumulative ReviewPage 445 Page 446
An
swer
s
xOf(x )
f (x) � x 2 � 2x � 8
(�1, �9)
(�4, 0) (2, 0)
x � �1
s � amount invested instock:
0.07s � 0.05(10,000 � s)� 550; at least $2500
consistent and dependent
A(4, �6), B(7, 0), C(1, �2)
2 or 0; 2 or 0; 4, 2, or 0
© Glencoe/McGraw-Hill A38 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. 12.
13. 14.
15.
16.
17. DCBA
DCBA
DCBA
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 5.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
9
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
5 9 6/
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
5 / 8
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
Chapter 7 Assessment Answer KeyStandardized Test Practice
Page 447 Page 448
© Glencoe/McGraw-Hill A39 Glencoe Algebra 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
1.
2.
3.
4.
5.
6.
7.
8. A
A
D
B
C
D
A
D
y
xO
f�1 (x) � �x �
72
�
51; 57
�1, �3, �5, �15,
��13
�, ��53
�
3 or 1; 0; 2 or 0
x � 2; x � 5
�86
10, �10, �2�, ��2�
Sample answer: rel.max. at x � �2 and x � 1, rel. min. at x � 0
xO
f(x )
�216
y
xO
y � �4(x � 1)2 � 3
(�3, �5); x � �3; up
0; 1 real root
��1 �
12i�95��
4x2 � 13x � 12 � 0
{�4, 6}
between �1 and 0; 2
75 m
xO
(2, 1)
(0, �3)
f(x )
f (x ) � –x 2 � 4x – 3x � 2
�2 � x � 1
�9m7�
2x2 � 3x � 4 � �x �
31
�
��110� � �
170�i
7�3� � 12�2�
4 � x � y2
�4y2
x�
�6y
5� 9
�
4x2 � 12x � 9
245x5
8x2 � 5x � 6
Chapter 7 Assessment Answer Key Unit 2 Test Semester TestPage 449 Page 450 Page 451
An
swer
s
(continued on the next page)
© Glencoe/McGraw-Hill A40 Glencoe Algebra 2
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.y
xO
D: x � ��52
�; R: y � 0
p�1(x) � �x �
48
�
�3x2 � 4x � 9;12x3 � 27x2
�6, �4i, 4i
y � �(x � 2)2 � 3
��1, �17
��
{�9, 7}
minimum; �14
6.596
2x2 � 7xy � 15y2
72x4y3
25
40 cars
c � 2t; 5c � 8t � 360;
5c � 8t � 10v � 410
(2, �5)
xO
f(x )
D � all reals; R � {y � y � 3}
��14
�
4a2 � 8a � 12
3 40 1 2�1
�m � ��12
� � m � 4�or ���
12
�, 4�
p � the number ofsheets of plywood; 270 � 7p � 1000; no
more than 104 sheets ofplywood
1256 cm2
C
B
B
A
C
A
D
B
B
C
Chapter 7 Assessment Answer Key Semester Test (continued)Page 452 Page 453 Page 454
� ��3 4�5 0
� �3 �61�7 15
�112�� �1 �2
2 8