polynomial vs. non-polynomial functions even vs. odd ... · 21. describe the degree and leading...

CC Algebra II HW #22

Polynomial vs. Non-Polynomial Functions Even vs. Odd Functions; End Behavior

Read 4.1 Examples 1-3 Section 4.1 2. Which One Doesn't Belong? Which function does not belong with the other three? Explain your reasoning.

In Exercises 3–8, decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. (See Example 1.) 3. 2653)( 23 +−+−= xxxxf 5. xxxxxf 2689)( 234 +−+= − 7. xxxxxh +−+−= 2

134235 87)(

9. Describe and correct the error in analyzing the function 113978)( 243 +−−−= xxxxxf .

11. Evaluate the function for the given value of x. (See Example 2.) 2 ;61223)( 34 −=−−+−= xxxxxh

In Exercises 17-20, describe the end behavior of the graph of the function. (See Example 3.) 17. 29675)( 234 ++−+−= xxxxxh 19. 284 1517122)( xxxxf +++−=

Name ____________________________ Period ___ Row___ Date ___________

21. Describe the degree and leading coefficient of the polynomial function using the graph.

23. Using Structure Determine whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient.

414295)( 55243253 −−−++−+= − xxxxxxxxxf

Section 4.8 Determine whether the function is even, odd, or neither. (See Example 4.)

41. 23)( 24 −+= xxxf 43. 15)( 2 ++= xxxg 44. 92)( 3 −+−= xxxf

CC Algebra II

HW #23

Adding, Subtracting & Multiplying Polynomials

Section 4.1 Analyzing Relationships In Exercises 33–36, describe the x-values for which (a) f is increasing or decreasing, (b) 0)( >xf , and (c) 0)( <xf .

Section 4.2 5. Find the sum. (See Example 1.) )1438()52312( 3445 ++−+−+− xxxxxx In Examples 9-14, find the difference. (See Example 2.) 9. )43125()8423( 2323 −−+−−+− xxxxxx 13. )4139()10268( 235235 +−−−+−+ xxxxxxx 15. Modeling With Mathematics During a recent period of time, the numbers (in thousands) of males M and females F

that attend degree-granting institutions in the United States can be modeled by

8.1012736828

6.75395.3107.192

2

++=

++=

ttF

ttM

where t is time in years.

Write a polynomial to model the total number of people attending degree-granting institutions. Interpret its constant term.


In Exercises 17–24, find the product. (See Example 3.) 17. )135(7 23 ++ xxx 19. )32)(645( 2 +−+− xxx 23. )12)(793( 23 +−+− xxxx 25. Error Analysis: Describe and correct the error in performing the operation.

In Exercises 27–32, find the product of the binomials. (See Example 4.) 27. )4)(2)(3( ++− xxx

CC Algebra II Name _________________________ HW #24 Period ___ Row ____ Date ______

Factoring with GCF, Difference of Two Squares, By Grouping, & Trinomials (a=1) Read 4.4 / Examples 1 & 3

Section 4.4 In Exercises 5–12, factor the polynomial completely. (See Example 1.) 5. xxx 242 23 −− 7. 35 1923 pp −

6. 35 1004 kk − 8. 456 64242 mmm +−

21. Error Analysis Describe and correct the error in factoring the polynomial. In Exercises 23–30, factor the polynomial completely. (See Example 3.) 23. 3065 23 −+− yyy 27. 3248 23 +−− xxx

25. 488183 23 +++ aaa 29. 369164 23 +−− qqq

CC Algebra II Name _________________________ HW #25 Period ___ Row ____ Date ______

Factoring with Sum/Difference of Cubes, Quadratic Form, & Trinomials (a > 1) Read 4.4 / Examples 2 & 4

Section 4.4 In Exercises 5–12, factor the polynomial completely. (See Example 1.) 9. 234 1892 qqq −+

11. 8910 61910 www +−

In Exercises 13–20, factor the polynomial completely. (See Example 2.) 13. 643 +x 17. 69 1923 hh −

15. 3433 −g 19. 47 25016 tt +

In Exercises 31–38, factor the polynomial completely. (See Example 4.) 31. 49k4 − 9 35. 16z4 − 81 33. c4 + 9c2 + 20 37. 3r8 + 3r5 − 60r2


Dividing Polynomials Including Synthetic Division Read 4.3 Examples 1-3

Section 4.3 Divide using polynomial long division. (See Example 1.) 5. )4()17( 2 −÷−+ xxx 7. )1()2( 223 −÷+++ xxxx In Exercises 11–18, divide using synthetic division. (See Examples 2 and 3.) 11. )4()18( 2 −÷++ xxx 13. )5()72( 2 +÷+− xxx 15. )3()9( 2 −÷+ xx 17. )6()121385( 234 −÷−+−− xxxxx


Error Analysis Describe and correct the error in using synthetic division to divide x3 − 5x + 3 by x − 2.

Maintaining Mathematical Proficiency Find the zero(s) of the function. (Sections 3.1 and 3.2) 41. 96)( 2 +−= xxxf 43. 4914)( 2 ++= xxxg


Remainder Theorem Read 4.3 Example 4

Section 4.3 6. Divide using polynomial long division. (See Example 1.) )5()5143( 2 −÷−− xxx In Exercises 25–32, use synthetic division to evaluate the function for the indicated value of x. (See Example 4.) 25. 1 ;308)( 2 −=+−−= xxxxf 27. 2 ;342)( 23 =++−= xxxxxf 29. 6 ;16)( 3 =+−= xxxxf 31. 3 ;176)( 24 =+−+= xxxxxf 33. Making an Argument You use synthetic division to divide )(xf by (x − a) and find that the remainder equals 15. Your friend concludes that af =)15( . Is your friend correct? Explain your reasoning.


34. Thought Provoking A polygon has an area represented by A = 4x2 + 8x + 4. The figure has at least one dimension equal to 2x + 2. Draw the figure and label its dimensions.


The Factor Theorem Read 4.4 Examples 5-7

Section 4.4 In Exercises 39–44, determine whether the binomial is a factor of the polynomial function. (See Example 5.) 39. 4 ;603752)( 23 −−−+= xxxxxf 41. 3 ;9156)( 345 +−−= xxxxxh In Exercises 45–50, show that the binomial is a factor of the polynomial. Then factor the function completely. (See Example 6.) 46. 5 ;4595)( 23 −+−−= xxxxxt 47. 6 ;4886)( 34 −+−−= xxxxxf 49. 7 ;8437)( 3 ++−= xxxxr


Analyzing Relationships In Exercises 51–54, match the function with the correct graph. Explain your reasoning. 51. )1)(3)(2()( +−−= xxxxf 52. )2)(1)(2()( −++= xxxxxg

53. )1)(3)(2()( −++= xxxxh 54. )2)(1)(2()( +−−= xxxxxk

69. Comparing Methods You are taking a test where calculators are not permitted. One question asks you to evaluate g(7) for the function g(x) = x3 − 7x2 − 4x + 28. You use the Factor Theorem and synthetic division and your friend uses direct substitution. Whose method do you prefer? Explain your reasoning. Maintaining Mathematical Proficiency Solve the quadratic equation by factoring. (Section 3.1) 79. 3x2 −11x + 10 = 0 Solve the quadratic equation by completing the square. (Section 3.3) 83. 3x2 + 30x + 63 = 0

CC Algebra II HW # 29

Sketching Polynomial Functions Using Zeros and End Behavior Read 4.5 Examples 1, 2 and 5; 4.6 Example 1

Section 4.5 In Exercises 3–12, solve the equation. (See Example 1.) 3. 01223 =−− zzz 9. cccc 361262 234 −=− In Exercises 13–20, find the zeros of the function. Then sketch a graph of the function. (See Example 2.) 13. 234 6)( xxxxh −+= 17. 234 6084)( xxxxg ++−= 19. 99)( 23 ++−−= xxxxh

In Exercises 41–46, write a polynomial function f of least degree that has a leading coefficient of 1 and the given zeros. (See Example 5.) 41. –2, 3, 6 45. 53 ,0 ,6 −−


Section 4.6 In Exercises 3–8, identify the number of solutions or zeros. (See Example 1.) 3. 042 234 =+−+ xxxx 7. 224)( 735 −+−= ssssg In Exercises 17–20, determine the number of imaginary zeros for the function with the given degree and graph. Explain your reasoning. 17. Degree: 4

18. Degree: 5

20. Degree: 3


Analyzing Graphs of Polynomial Functions Read 4.8 Examples 1 and 3

Section 4.8 1. Complete the Sentence A local maximum or local minimum of a polynomial function occurs at a _____________________ point of the graph of the function.

2. Writing Explain what a local maximum of a function is and how it may be different from the maximum value of the function. Analyzing Relationships In Exercises 3–6, match the function with its graph. 3. )2)(2)(1()( +−−= xxxxf 4. )1()2()( 2 ++= xxxh

5. )2)(1)(1()( +−+= xxxxg 6. )2()1()( 2 +−= xxxf

In Exercises 7–14, graph the function. (See Example 1: Be sure to use this method, and not just graph it on your calculator!) 7. )1()2()( 2 +−= xxxf 9. )3)(1()1()( 2 −−+= xxxxh

Error Analysis Describe and correct the error in using factors to graph f. 15. 2)1)(2()( −+= xxxf


In Exercises 23–30, graph the function. Identify the x-intercepts and the points where the local maximums and local minimums occur. Determine the intervals for which the function is increasing or decreasing. (See Example 3.) 23. 382)( 23 −+= xxxg 28. xxxxf 537.0)( 34 +−=

37. Open-Ended Sketch a graph of a polynomial function f having the given characteristics.

• The graph of f has x-intercepts at x = −4, x = 0, and x = 2. • f has a local maximum value when x = 1. • f has a local minimum value when x = −2.

47. Using Tools When a swimmer does the breaststroke, the function

tttttttS 43.2144737165018701060241 234567 −+−+−+−=

models the speed S (in meters per second) of the swimmer during one complete stroke, where t is the number of seconds since the start of the stroke and 0 ≤ t ≤ 1.22. Use a graphing calculator to graph the function. At what time during the stroke is the swimmer traveling the fastest?

polynomial vs. non-polynomial functions even vs. odd ... · 21. describe the degree and leading...

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