polynomials and polynomial...

� � � � � � � �POLYNOMIALS ANDPOLYNOMIALFUNCTIONS

POLYNOMIALS ANDPOLYNOMIALFUNCTIONS

c What type of function modelsthe speed of a space shuttle?

320

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APPLICATION: Space Exploration

The space shuttle’s engines produce immense power, equiva-

lent to the output of 23 Hoover

Dams. All this power is needed

to accelerate the shuttle to more

than 17,000 miles per hour in

about eight minutes after launch.

This speed allows the shuttle to

achieve and maintain an orbit

240 miles above Earth’s surface.

Think & Discuss

The table below gives the time (in seconds) afterlaunch and the corresponding average speed (in feetper second) of the shuttle.

1. Make a scatter plot of the data. Estimate how longit takes the shuttle to reach a speed of 1000 feet persecond.

2. Would either a linear function or a quadraticfunction be a good model for the data? Explain.

Learn More About It

You will model the speed of the space shuttle inExercise 49 on p. 385.

APPLICATION LINK Visit www.mcdougallittell.com

for more information on space exploration.

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C H A P T E R

6

321

c

Time (sec) Speed (ft/sec)

20 463.4

40 979.3

60 1421.3

80 2283.5

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322 Chapter 6

What’s the chapter about?

Chapter 6 is about polynomials, polynomial equations, and polynomial functions.In Chapter 6 you’ll learn

• how to perform operations on polynomials and solve polynomial equations.

• how to evaluate, graph, and find zeros of polynomial functions.

CHAPTER

6Study Guide

PREVIEW

Are you ready for the chapter?

SKILL REVIEW Do these exercises to review key skills that you’ll apply in thischapter. See the given reference page if there is something you don’t understand.

Simplify the expression. (Review Example 5, p. 13)

1. 4x2 º 2x + x º x2 2. 2(8x + 5) º 19x 3. ºx

3 º 5x4 º 3x

3 + 7x2

Graph the quadratic function. (Review Examples 1–3, pp. 250 and 251)

4. y = º3(x º 2)2 5. y = (x + 1)(x º 5) 6. y = 2(x + 6)(x + 4)

Write the quadratic function in standard form. (Review Example 4, p. 251)

7. y = (x º 1)2 º 7 8. y = 2(x + 4)2 9. y = º(x º 2)(x + 8)

Solve the equation. (Review Example 5, p. 258)

10. x2 + 6x º 27 = 0 11. x

2 + 20x + 100 = 0 12. 2x2 + 5x º 12 = 0

PREPARE

Here’s a study strategy!

STUDY

STRATEGY

c Review

• power, p. 11

• x-intercept, p. 84

• zeros of a function, p. 259

c New

• polynomial function, p. 329

• end behavior, p. 331

• polynomial long division, p. 352

• synthetic division, p. 353

• rational zero theorem, p. 359

• fundamental theorem ofalgebra, p. 366

• local maximum, p. 374

• local minimum, p. 374

• finite differences, p. 380

KEY VOCABULARY

STUDENT HELP

Study Tip“Student Help” boxesthroughout the chaptergive you study tips andtell you where to look forextra help in this bookand on the Internet.

Making a Flow ChartA flow chart is a diagram that shows the possiblepaths and steps you can follow to solve a problem.After you complete the chapter, make a flowchart that shows how to find all the zeros of apolynomial function. Include techniques andtheorems you learned in Chapter 6 which you canuse with various types of polynomial functions.

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6.1 Using Properties of Exponents 323

Using Properties of Exponents

PROPERTIES OF EXPONENTS

Recall that the expression an, where n is a positive integer, represents the product thatyou obtain when a is used as a factor n times. In the activity you will investigate twoproperties of exponents.

In the activity you may have discovered two of the following properties of exponents.

GOAL 1

Use properties of

exponents to evaluate and

simplify expressions involving

powers.

Use exponents and

scientific notation to solve

real-life problems, such as

finding the per capita GDP of

Denmark in Example 4.

. To simplify real-lifeexpressions, such as the

ratio of a state’s park

space to total area in

Ex. 57.

Why you should learn it

GOAL 2

GOAL 1

What you should learn

6.1

Products and Quotients of Powers

How many factors of 2 are there in the product 23 • 24? Use your answer towrite the product as a single power of 2.

Write each product as a single power of 2 by counting the factors of 2. Usea calculator to check your answers.

a. 22 • 25 b. 21 • 26 c. 23 • 26 d. 24 • 24

Complete this equation: 2m • 2n = 2?

Write each quotient as a single power of 2 by first writing the numeratorand denominator in “expanded form” (for example, 23 = 2 • 2 • 2) and thencanceling common factors. Use a calculator to check your answers.

a. b. c. d.

Complete this equation: = 2?2m

}

2n5

26

}

22

27

}

23

25

}

22

23

}

21

4

3

2

1

DevelopingConcepts

ACTIVITY

REA

L LIFE

REA

L LIFE

Let a and b be real numbers and let m and n be integers.

PRODUCT OF POWERS PROPERTY am • a

n = am + n

POWER OF A POWER PROPERTY (am)n = amn

POWER OF A PRODUCT PROPERTY (ab)m = amb

m

NEGATIVE EXPONENT PROPERTY aºm = , a ≠ 0

ZERO EXPONENT PROPERTY a0 = 1, a ≠ 0

QUOTIENT OF POWERS PROPERTY = am º n, a ≠ 0

POWER OF A QUOTIENT PROPERTY S}

b

a}Dm

= , b ≠ 0a

m

}

bm

am

}

an

1}

am

PROPERTIES OF EXPONENTSCONCEPT

SUMMARYLake Clark National Park,

Alaska

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324 Chapter 6 Polynomials and Polynomial Functions

The properties of exponents can be used to evaluate numerical expressions and tosimplify algebraic expressions. In this book we assume that any base with a zero ornegative exponent is nonzero. A simplified algebraic expression contains onlypositive exponents.

Evaluating Numerical Expressions

a. (23)4 = 23 • 4 Power of a power property

= 212 Simplify exponent.

= 4096 Evaluate power.

b. S}

34

}D2= Power of a quotient property

= }

196} Evaluate powers.

c. (º5)º6(º5)4 = (º5)º6 + 4 Product of powers property

= (º5)º2 Simplify exponent.

= }(º

1

5)2} Negative exponent property

= }

215} Evaluate power.

Simplifying Algebraic Expressions

a. S}

sº

r5

}D2= Power of a quotient property

= Power of a power property

= r2s10 Negative exponent property

b. (7bº3)2b5b = 72(bº3)2b5b Power of a product property

= 49bº6b5b Power of a power property

= 49bº6 + 5 + 1 Product of powers property

= 49b0 Simplify exponent.

= 49 Zero exponent property

c. = Power of a product property

= Power of a power property

= x2 º 3y4 º (º1) Quotient of powers property

= xº1y5 Simplify exponents.

= }

y

x

5

} Negative exponent property

x2y4

}

x3yº1

x2(y2)2

}

x3yº1

(xy2)2

}

x3yº1

r2

}

sº10

r2

}

(sº5)2

E X A M P L E 2

32

}

42

E X A M P L E 1

STUDENT HELP

Study Tip

When you multiplypowers, do not multiplythe bases. For example,23 • 25

≠ 48.

HOMEWORK HELP

Visit our Web sitewww.mcdougallittell.comfor extra examples.

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STUDENT HELP

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USING PROPERTIES OF EXPONENTS IN REAL LIFE

Comparing Real-Life Volumes

ASTRONOMY The radius of the sun is about 109 times as great as Earth’s radius.How many times as great as Earth’s volume is the sun’s volume?

SOLUTION

Let r represent Earth’s radius.

= The volume of a sphere is }43}πr 3.

= Power of a product property

= 1093r

0 Quotient of powers property

= 1093 Zero exponent property

= 1,295,029 Evaluate power.

c The sun’s volume is about 1.3 million times as great as Earth’s volume.

. . . . . . . . . .

A number is expressed in if it is in the form c ª 10n where1 ≤ c < 10 and n is an integer. For instance, the width of a molecule of water isabout 2.5 ª 10º8 meter, or 0.000000025 meter. When working with numbers inscientific notation, the properties of exponents listed on page 323 can help makecalculations easier.

Using Scientific Notation in Real Life

In 1997 Denmark had a population of 5,284,000 and a gross domestic product (GDP)of $131,400,000,000. Estimate the per capita GDP of Denmark.

DATA UPDATE of UN/ECE Statistical Division data at www.mcdougallittell.com

SOLUTION

“Per capita” means per person, so divide the GDP by the population.

}

PopGuDla

Ption

} = }131

5,4,20804,0,00000,000

} Divide GDP by population.

= Write in scientific notation.

= }15

.

.32

18

44

} ª 105 Quotient of powers property

≈ 0.249 ª 105 Use a calculator.

= 24,900 Write in standard notation.

c The per capita GDP of Denmark in 1997 was about $25,000 per person.

1.314 ª 1011

}}

5.284 ª 106

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E X A M P L E 4

scientific notation

}

43

}π1093r

3

}}

}

43

}πr3

}

43

}π(109r)3

}}

}

43

}πr3

Sun’s volume}}

Earth’s volume

E X A M P L E 3

GOAL 2

STUDENT HELP

Skills Review For help with scientificnotation, see p. 913.

ASTRONOMY

Jupiter is the largestplanet in the solar system. Ithas a radius of 71,400 km—over 11 times as great asEarth’s, but only about onetenth as great as the sun’s.

APPLICATION LINK

www.mcdougallittell.com

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FOCUS ON

APPLICATIONS

Earth

Sun

Jupiter

RE

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Economics

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1. State the name of the property illustrated.

a. am • an = am + n b. (am)n = amn c. (ab)m = ambm

2. ERROR ANALYSIS Describe the mistake made in simplifying the expression.

a. b. c.

Evaluate the expression. Tell which properties of exponents you used.

3. 6 • 62 4. (96)(92)º3 5. (23)2

6. S DS D27. S}

35

}Dº28. }

7

7º

º

3

5

}

Simplify the expression. Tell which properties of exponents you used.

9. zº2 • zº4 • z6 10. yzº2(x2y)3z 11. (4x3)º2

12. S D613. 14.

15. ASTRONOMY Earth has a radius of about 6.38 ª 103 kilometers. The sunhas a radius of about 6.96 ª 105 kilometers. Use the formula for the volume of asphere given on page 325 to calculate the volume of the sun and the volume ofEarth. Divide the volumes. Do you get the same result as in Example 3?

EVALUATING NUMERICAL EXPRESSIONS Evaluate the expression. Tell which

properties of exponents you used.

16. 42 • 44 17. (5º2)3 18. (º9)(º9)3 19. (82)3

20. 21. S}

37

}D322. S}

59

}Dº323. 11º2 • 110

24. 25. S}

18

}Dº426. (2º4)º2 27.

28. 29. 60 • 63 • 6º4 30. S}

110}D3S}

110}Dº3

31. SS}

25

}Dº3D2

SIMPLIFYING ALGEBRAIC EXPRESSIONS Simplify the expression. Tell which

properties of exponents you used.

32. x8 • }

x

13} 33. (23x2)5 34. (x2y2)º1 35. }

x

xº

5

2}

36. 37. (x4y7)º3 38. 39. º3xº4y0

40. (10x3y5)º3 41. 42. (4x2y5)º2 43.

44. 45. • 46. • }

2

x

0

y

x6

14

} 47. •7x5y2

}

4y

12xy}

7x4

y10

}

2x3

º7y}

21x5

xy9

}

3yº2

5x3y9

}

20x2yº2

2x2y}

6xyº1

xº1y}

xyº2

x11y10

}

xº3yº1

x5y2

}

x4y0

62

}

(6º2 • 51)º2

22

}

2º9

4º2

}

4º3

52

}

55

PRACTICE AND APPLICATIONS

(xy)4

}

xyº1

3y6

}

y3

2}

xº3

1}

23

}

2º2

GUIDED PRACTICE

Vocabulary Check ✓

Concept Check ✓

Skill Check ✓

STUDENT HELP

Extra Practice to help you masterskills is on p. 947.

(º2)2 (º2)3 = 45}

x

x2

8

} = x4

x4 • x3 = x12

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 16–31Example 2: Exs. 32–51Examples 3, 4: Exs. 52–56

! " # $ % & 'Write an expression for the area or volume of the

figure in terms of x.

48. A = s2 49. A = πr2

50. V = πr2h 51. V = }

43

}πr3

SCIENTIFIC NOTATION In Exercises 52–56, use scientific notation.

52. NATIONAL DEBT On June 8, 1999, the national debt of the United States was about $5,608,000,000,000. The population of the United States at that time was about 273,000,000. Suppose the national debt was divided evenly amongeveryone in the United States. How much would each person owe?

DATA UPDATE of Bureau of the Public Debt and U.S. Census Bureau data at www.mcdougallittell.com

53. The table shows the population and grossdomestic product (GDP) in 1997 for each of six different countries. Calculate the per capita GDP for each country.

DATA UPDATE of UN/ECE Statistical Division data at www.mcdougallittell.com

54. A red blood cell has a diameter of approximately

0.00075 centimeter. Suppose one of the arteries in your body has a diameter of0.0456 centimeter. How many red blood cells could fit across the artery?

55. SPACE EXPLORATION On February 17, 1998, Voyager 1 became the mostdistant manmade object in space, at a distance of 10,400,000,000 kilometersfrom Earth. How long did it take Voyager 1 to travel this distance given that ittraveled an average of 1,390,000 kilometers per day? c Source: NASA

56. ORNITHOLOGY Some scientists estimate that there are about 8600 speciesof birds in the world. The mean number of birds per species is approximately12,000,000. About how many birds are there in the world?

BIOLOGY CONNECTION

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SOCIAL STUDIES CONNECTION

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Ï3w}

4

GEOMETRY CONNECTION


x2

2x

x

4x

x3

Country Population GDP (U.S. dollars)

France 58,607,000 1,249,600,000,000

Germany 82,061,000 1,839,300,000,000

Ireland 3,661,000 71,300,000,000

Luxembourg 420,000 13,600,000,000

The Netherlands 15,600,000 333,400,000,000

Sweden 8,849,000 177,300,000,000

ORNITHOLOGIST

An ornithologist is ascientist who studies thehistory, classification, bio-logy, and behavior of birds.

CAREER LINK


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FOCUS ON

CAREERS

( ) * + , - . /


57. MULTI-STEP PROBLEM Suppose you live in a state that has a total area of 5.38 ª 107 acres and 4.19 ª 105 acres of park space. You think that the stateshould set aside more land for parks. The table shows the total area and theamount of park space for several states.

a. Write the total area and the amount of park space for each state in scientificnotation.

b. For each state, divide the amount of park space by the total area.

c. Writing You want to ask the state legislature to increase the amount of parkspace in your state. Use your results from parts (a) and (b) to write a letter thatexplains why your state needs more park space.

LOGICAL REASONING In Exercises 58 and 59, refer to the properties of

exponents on page 323.

58. Show how the negative exponent property can be derived from the quotient ofpowers property and the zero exponent property.

59. Show how the quotient of powers property can be derived from the product ofpowers property and the negative exponent property.

GRAPHING Graph the equation. (Review 2.3, 5.1 for 6.2)

60. y = º4 61. y = ºx º 3 62. y = 3x + 1

63. y = º2x + 5 64. y = 3x2 + 2 65. y = º2x(x + 6)

66. y = x2 º 2x º 6 67. y = 2x2 º 4x + 10 68. y = º2(x º 3)2 + 8

SOLVING QUADRATIC EQUATIONS Solve the equation. (Review 5.3)

69. 2x2 = 32 70. º3x2 = º24 71. 25x2 = 16

72. 3x2 º 8 = 100 73. 13 º 5x2 = 8 74. 4x2 º 5 = 9

75. ºx2 + 9 = 2x2 º 6 76. 12 + 2x2 = 5x2 º 8 77. º2x2 + 7 = x2 º 2

OPERATIONS WITH COMPLEX NUMBERS Write the expression as a complex

number in standard form. (Review 5.4)

78. (9 + 4i) + (9 º i) 79. (º5 + 3i) º (º2 º i) 80. (10 º i) º (4 + 7i)

81. ºi(7 + 2i) 82. º11i(5 + i) 83. (3 + i)(9 + i)

MIXED REVIEW

TestPreparation

★★Challenge

State Total area (acres) Amount of park space (acres)

Alaska 393,747,200 3,250,000

California 101,676,000 1,345,000

Connecticut 3,548,000 176,000

Kansas 52,660,000 29,000

Ohio 28,690,000 204,000

Pennsylvania 29,477,000 283,000

c Source: Statistical Abstract of the United States

EXTRA CHALLENGE


( ) * + , - . /Evaluating and GraphingPolynomial Functions

EVALUATING POLYNOMIAL FUNCTIONS

A is a function of the form

ƒ(x) = anxn + an º 1xn º 1 + . . . + a1x + a0

where an ≠ 0, the exponents are all whole numbers, and the coefficients are all realnumbers. For this polynomial function, an is the a0 is the

and n is the A polynomial function is in if its terms are written in descending order of exponents from left to right.

You are already familiar with some types of polynomial functions. For instance, thelinear function ƒ(x) = 3x + 2 is a polynomial function of degree 1. The quadraticfunction ƒ(x) = x2 + 3x + 2 is a polynomial function of degree 2. Here is a summaryof common types of polynomial functions.

Identifying Polynomial Functions

Decide whether the function is a polynomial function. If it is, write the function instandard form and state its degree, type, and leading coefficient.

a. ƒ(x) = }12

}x2 º 3x4 º 7 b. ƒ(x) = x3 + 3x

c. ƒ(x) = 6x2 + 2xº1 + x d. ƒ(x) = º0.5x + πx2 º Ï2w

SOLUTION

a. The function is a polynomial function. Its standard form is ƒ(x) = º3x4 + }21

}x2 º 7.It has degree 4, so it is a quartic function. The leading coefficient is º3.

b. The function is not a polynomial function because the term 3x does not have avariable base and an exponent that is a whole number.

c. The function is not a polynomial function because the term 2xº1 has an exponent thatis not a whole number.

d. The function is a polynomial function. Its standard form is ƒ(x) = πx2 º 0.5x º Ï2w.It has degree 2, so it is a quadratic function. The leading coefficient is π.

E X A M P L E 1

standard formdegree.constant term,leading coefficient,

polynomial function

GOAL 1

6.2 Evaluating and Graphing Polynomial Functions 329

Evaluate a

polynomial function.

Graph a polynomial

function, as applied in

Example 5.

. To find values of real-lifefunctions, such as the

amount of prize money

awarded at the U.S. Open

Tennis Tournament in

Ex. 86.


GOAL 2

GOAL 1


6.2R

EAL LIFE

REA

L LIFE Degree Type Standard form

0 Constant ƒ(x) = a0

1 Linear ƒ(x) = a1x + a0

2 Quadratic ƒ(x) = a2x2 + a1x + a0

3 Cubic ƒ(x) = a3x3 + a2x2 + a1x + a0

4 Quartic ƒ(x) = a4x4 + a3x3 + a2x2 + a1x + a0

0 1 2 3 4 5 6 7


One way to evaluate a polynomial function is to use direct substitution. For instance,ƒ(x) = 2x4 º 8x2 + 5x º 7 can be evaluated when x = 3 as follows.

ƒ(3) = 2(3)4 º 8(3)2 + 5(3) º 7

= 162 º 72 + 15 º 7

= 98

Another way to evaluate a polynomial function is to use

Using Synthetic Substitution

Use synthetic substitution to evaluate ƒ(x) = 2x4 º 8x2 + 5x º 7 when x = 3.

SOLUTION

Write the value of x and the coefficients of ƒ(x) as shown. Bring down the leadingcoefficient. Multiply by 3 and write the result in the next column. Add the numbers inthat column and write the sum below the line. Continue to multiply and add, as shown.

c ƒ(3) = 98

. . . . . . . . . .

Using synthetic substitution is equivalent to evaluating the polynomial in nested form.

ƒ(x) = 2x4 + 0x3 º 8x2 + 5x º 7 Write original function.

= (2x3 + 0x2 º 8x + 5)x º 7 Factor x out of first 4 terms.

= ((2x2 + 0x º 8)x + 5)x º 7 Factor x out of first 3 terms.

= (((2x + 0)x º 8)x + 5)x º 7 Factor x out of first 2 terms.

Evaluating a Polynomial Function in Real Life

PHOTOGRAPHY The time t (in seconds) it takes a camera battery to recharge afterflashing n times can be modeled by t = 0.000015n3 º 0.0034n2 + 0.25n + 5.3. Findthe recharge time after 100 flashes. c Source: Popular Photography

SOLUTION

c The recharge time is about 11 seconds.

E X A M P L E 3

E X A M P L E 2

synthetic substitution.

100 0.000015 º0.0034 0.25 5.30.0015 º0.19 6

0.000015 º0.0019 0.06 11.3

PHOTOGRAPHER

Some photographerswork in advertising, somework for newspapers, andsome are self-employed.Others specialize in aerial,police, medical, or scientificphotography.

CAREER LINK


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x-value

2x4 + 0x3 + (º8x2) + 5x + (º7) Polynomial in standard form

3 2 0 º8 5 º7 Coefficients

6 18 30 105

2 6 10 35 98The value of ƒ(3) is the lastnumber you write, in thebottom right-hand corner.

Study TipIn Example 2, note thatthe row of coefficients for ƒ(x) must include acoefficient of 0 for the“missing” x3-term.

STUDENT HELP

0 1 2 3 4 5 6 7GRAPHING POLYNOMIAL FUNCTIONS

The of a polynomial function’s graph is the behavior of the graph as xapproaches positive infinity (+‡) or negative infinity (º‡). The expression x ˘ +‡is read as “x approaches positive infinity.”

f (x) → 2`

as x → 2`.f (x) → 2`

as x → 1`.

8 99f (x) → 1`

as x → 2`.f (x) → 1`

as x → 1`.8f (x) → 1` as x → 2`.

f (x) → 2` as x → 1`.

98f (x) → 1` as x → 1`.

f (x) → 2` as x → 2`.

8 9end behavior

GOAL 2

Investigating End Behavior

Use a graphing calculator to graph each function. Then complete thesestatements: ƒ(x) ˘

ooo

? as x ˘ º‡ and ƒ(x) ˘ ooo

? as x ˘ +‡.

a. ƒ(x) = x3 b. ƒ(x) = x4 c. ƒ(x) = x5 d. ƒ(x) = x6

e. ƒ(x) = ºx3 f. ƒ(x) = ºx4 g. ƒ(x) = ºx5 h. ƒ(x) = ºx6

How does the sign of the leading coefficient affect the behavior of apolynomial function’s graph as x ˘ +‡?

How is the behavior of a polynomial function’s graph as x ˘ +‡ related toits behavior as x ˘ º‡ when the function’s degree is odd? when it is even?

3

2

1

DevelopingConcepts

ACTIVITY

The graph of ƒ(x) = an x n + an º 1xn º 1 + . . . + a1x + a0 has this end behavior:

• For an

> 0 and n even, ƒ(x) ˘ +‡ as x ˘ º‡ and ƒ(x) ˘ +‡ as x ˘ +‡.

• For an

> 0 and n odd, ƒ(x) ˘ º‡ as x ˘ º‡ and ƒ(x) ˘ +‡ as x ˘ +‡.

• For an

< 0 and n even, ƒ(x) ˘ º‡ as x ˘ º‡ and ƒ(x) ˘ º‡ as x ˘ +‡.

• For an

< 0 and n odd, ƒ(x) ˘ +‡ as x ˘ º‡ and ƒ(x) ˘ º‡ as x ˘ +‡.

END BEHAVIOR FOR POLYNOMIAL FUNCTIONSCONCEPT

SUMMARY

In the activity you may have discovered that the end behavior of a polynomialfunction’s graph is determined by the function’s degree and leading coefficient.


STUDENT HELP

Look Back For help with graphingfunctions, see pp. 69and 250.

: ; < = > ? @ AGraphing Polynomial Functions

Graph (a) ƒ(x) = x3 + x2 º 4x º 1 and (b) ƒ(x) = ºx4 º 2x3 + 2x2 + 4x.

SOLUTION

a. To graph the function, make a table of values and plot thecorresponding points. Connect the points with a smooth curve and check the end behavior.

The degree is odd and the leading coefficient is positive, so ƒ(x) ˘ º‡ as x ˘ º‡ and ƒ(x) ˘ +‡ as x ˘ +‡.

b. To graph the function, make a table of values and plot thecorresponding points. Connect the points with a smooth curve and check the end behavior.

The degree is even and the leading coefficient is negative, so ƒ(x) ˘ º‡ as x ˘ º‡ and ƒ(x) ˘ º‡ as x ˘ +‡.

Graphing a Polynomial Model

A rainbow trout can grow up to 40 inches in length. The weight y (in pounds) of a rainbow trout is related to its length x (in inches) according to the modely = 0.0005x3. Graph the model. Use your graph to estimate the length of a 10 pound rainbow trout.

SOLUTION

Make a table of values. The model makes sense only for positive values of x.

Plot the points and connect them with asmooth curve, as shown at the right. Noticethat the leading coefficient of the model ispositive and the degree is odd, so the graphrises to the right.

Read the graph backwards to see that x ≈ 27when y = 10.

c A 10 pound trout is approximately27 inches long.

E X A M P L E 5

E X A M P L E 4


8 9BC8 9BB

Size of Rainbow Trout

Length (in.)

We

igh

t (l

b)

0

810 20 30 40

0

30

10

20 9

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AL LIFE

RE

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Biology

x º3 º2 º1 0 1 2 3

ƒ(x) º7 3 3 º1 º3 3 23

x º3 º2 º1 0 1 2 3

ƒ(x) º21 0 º1 0 3 º16 º105

x 0 5 10 15 20 25 30 35 40

y 0 0.0625 0.5 1.69 4 7.81 13.5 21.4 32

HOMEWORK HELP


INT

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STUDENT HELP

: ; < = > ? @ A


1. Identify the degree, type, leading coefficient, and constant term of thepolynomial function ƒ(x) = 5x º 2x3.

2. Complete the synthetic substitution shown atthe right. Describe each step of the process.

3. Describe the graph of a constant function.

Decide whether each function is a polynomial function. If it is, use synthetic

substitution to evaluate the function when x = º1.

4. ƒ(x) = x4 Ï5w º x 5. ƒ(x) = x3 + x2 º xº3 + 3

6. ƒ(x) = 62x º 12x 7. ƒ(x) = 14 º 21x2 + 5x4

Describe the end behavior of the graph of the polynomial function by

completing the statements ƒ(x) ˘ ooo

? as x ˘ º‡ and ƒ(x) ˘ ooo

? as x ˘ +‡.

8. ƒ(x) = x3 º 5x 9. ƒ(x) = ºx5 º 3x3 + 2 10. ƒ(x) = x4 º 4x2 + x

11. ƒ(x) = x + 12 12. ƒ(x) = ºx2 + 3x + 1 13. ƒ(x) = ºx8 + 9x5 º 2x4

14. VIDEO RENTALS The total revenue (actual and projected) from home videorentals in the United States from 1985 to 2005 can be modeled by

R = 1.8t3 º 76t2 + 1099t + 2600

where R is the revenue (in millions of dollars) and t is the number of years since1985. Graph the function. c Source: The Wall Street Journal Almanac

CLASSIFYING POLYNOMIALS Decide whether the function is a polynomial

function. If it is, write the function in standard form and state the degree, type,

and leading coefficient.

15. ƒ(x) = 12 º 5x 16. ƒ(x) = 2x + }35

}x4 + 9 17. ƒ(x) = x + π

18. ƒ(x) = x2 Ï2w + x º 5 19. ƒ(x) = x º 3xº2 º 2x3 20. ƒ(x) = º2

21. ƒ(x) = x2 º x + 1 22. ƒ(x) = 22 º 19x + 2x 23. ƒ(x) = 36x2 º x3 + x4

24. ƒ(x) = 3x2 º 2xºx 25. ƒ(x) = 3x3 26. ƒ(x) = º6x2 + x º

DIRECT SUBSTITUTION Use direct substitution to evaluate the polynomial

function for the given value of x.

27. ƒ(x) = 2x3 + 5x2 + 4x + 8, x = º2 28. ƒ(x) = 2x3 º x4 + 5x2 º x, x = 3

29. ƒ(x) = x + }12

}x3, x = 4 30. ƒ(x) = x2 º x5 + 1, x = º1

31. ƒ(x) = 5x4 º 8x3 + 7x2, x = 1 32. ƒ(x) = x3 + 3x2 º 2x + 5, x = º3

33. ƒ(x) = 11x3 º 6x2 + 2, x = 0 34. ƒ(x) = x4 º 2x + 7, x = 2

35. ƒ(x) = 7x3 + 9x2 + 3x, x = 10 36. ƒ(x) = ºx5 º 4x3 + 6x2 º x, x = º2

3}x


GUIDED PRACTICE

º2 3 1 º9 2? ? ?

3 ? ? 0


Concept Check ✓

Skill Check ✓

STUDENT HELP


STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 15–26Example 2: Exs. 37–46Example 3: Exs. 81, 82Example 4: Exs. 47–79Example 5: Exs. 83–86

D E F G H I J KSYNTHETIC SUBSTITUTION Use synthetic substitution to evaluate the

polynomial function for the given value of x.

37. ƒ(x) = 5x3 + 4x2 + 8x + 1, x = 2 38. ƒ(x) = º3x3 + 7x2 º 4x + 8, x = 3

39. ƒ(x) = x3 + 3x2 + 6x º 11, x = º5 40. ƒ(x) = x3 º x2 + 12x + 15, x = º1

41. ƒ(x) = º4x3 + 3x º 5, x = 2 42. ƒ(x) = ºx4 + x3 º x + 1, x = º3

43. ƒ(x) = 2x4 + x3 º 3x2 + 5x, x = º1 44. ƒ(x) = 3x5 º 2x2 + x, x = 2

45. ƒ(x) = 2x3 º x2 + 6x, x = 5 46. ƒ(x) = ºx4 + 8x3 + 13x º 4, x = º2

END BEHAVIOR PATTERNS Graph each polynomial function in the table.

Then copy and complete the table to describe the end behavior of the graph

of each function.

47. 48.

MATCHING Use what you know about end behavior to match the polynomial

function with its graph.

49. ƒ(x) = 4x6 º 3x2 + 5x º 2 50. ƒ(x) = º2x3 + 5x2

51. ƒ(x) = ºx4 + 1 52. ƒ(x) = 6x3 + 1

A. B.

C. D.

DESCRIBING END BEHAVIOR Describe the end behavior of the graph of the

polynomial function by completing these statements: ƒ(x) ˘ooo

? as x ˘ º‡ and

ƒ(x) ˘ooo

? as x ˘ +‡.

53. ƒ(x) = º5x4 54. ƒ(x) = ºx2 + 1 55. ƒ(x) = 2x

56. ƒ(x) = º10x3 57. ƒ(x) = ºx6 + 2x3 º x 58. ƒ(x) = x5 + 2x2

59. ƒ(x) = º3x5 º 4x2 + 3 60. ƒ(x) = x7 º 3x3 + 2x 61. ƒ(x) = 3x6 º x º 4

62. ƒ(x) = 3x8 º 4x3 63. ƒ(x) = º6x3 + 10x 64. ƒ(x) = x4 º 5x3 + x º 1

LMML

L ML M


As AsFunction

x ˘ º‡ x ˘ +‡

ƒ(x) = º5x 3 ? ?

ƒ(x) = ºx 3 + 1 ? ?

ƒ(x) = 2x º 3x 3 ? ?

ƒ(x) = 2x 2 º x 3 ? ?

As AsFunction

x ˘ º‡ x ˘ +‡

ƒ(x) = x 4 + 3x 3 ? ?

ƒ(x) = x 4 + 2 ? ?

ƒ(x) = x 4 º 2x º 1 ? ?

ƒ(x) = 3x 4 º 5x 2 ? ?

D E F G H I J K


GRAPHING POLYNOMIALS Graph the polynomial function.

65. ƒ(x) = ºx3 66. ƒ(x) = ºx4 67. ƒ(x) = x5 + 2

68. ƒ(x) = x4 º 4 69. ƒ(x) = x4 + 6x2 º 5 70. ƒ(x) = 2 º x3

71. ƒ(x) = x5 º 2 72. ƒ(x) = ºx4 + 3 73. ƒ(x) = ºx3 + 3x

74. ƒ(x) = ºx3 + 2x2 º 4 75. ƒ(x) = ºx5 + x2 + 1 76. ƒ(x) = x3 º 3x º 1

77. ƒ(x) = x5 + 3x3 º x 78. ƒ(x) = x4 º 2x º 3 79. ƒ(x) = ºx4 + 2x º 1

80. CRITICAL THINKING Give an example of a polynomial function ƒ such thatƒ(x) ˘ º‡ as x ˘ º‡ and ƒ(x) ˘ +‡ as x ˘ +‡.

81. SHOPPING The retail space in shopping centers in the United States from1972 to 1996 can be modeled by

S = º0.0068t3 º 0.27t2 + 150t + 1700

where S is the amount of retail space (in millions of square feet) and t is thenumber of years since 1972. How much retail space was there in 1990?

82. CABLE TELEVISION The average monthly cable TV rate from 1980 to1997 can be modeled by

R = º0.0036t3 + 0.13t2 º 0.073t + 7.7

where R is the monthly rate (in dollars) and t is the number of years since 1980.What was the monthly rate in 1983?

NURSING In Exercises 83 and 84, use the following information.

From 1985 to 1995, the number of graduates from nursing schools in the UnitedStates can be modeled by

y = º0.036t4 + 0.605t3 º 1.87t2 º 4.67t + 82.5

where y is the number of graduates (in thousands) and t is the number of years since 1985. c Source: Statistical Abstract of the United States

83. Describe the end behavior of the graph of the function. From the end behavior,would you expect the number of nursing graduates in the year 2010 to be morethan or less than the number of nursing graduates in 1995? Explain.

84. Graph the function for 0 ≤ t ≤ 10. Use the graph to find the first year in whichthere were over 82,500 nursing graduates.

TENNIS In Exercises 85 and 86, use the following information.

The amount of prize money for the women’s U.S. Open Tennis Tournament from1970 to 1997 can be modeled by

P = 1.141t2 º 5.837t + 14.31

where P is the prize money (in thousands of dollars) and t is the number of yearssince 1970. c Source: U.S. Open

85. Describe the end behavior of the graph of the function. From the end behavior,would you expect the amount of prize money in the year 2005 to be more than orless than the amount in 1995? Explain.

86. Graph the function for 0 ≤ t ≤ 40. Use the graph to estimate the amount of prizemoney in the year 2005.

NURSE

Although themajority of nurses work inhospitals, nurses also workin doctors’ offices, in privatehomes, at nursing homes,and in other communitysettings.

CAREER LINK


INT

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FOCUS ON

CAREERS

D E F G H I J K87. MULTI-STEP PROBLEM To determine whether a Holstein heifer’s height is

normal, a veterinarian can use the cubic functions

L = 0.0007t3 º 0.061t2 + 2.02t + 30

H = 0.001t3 º 0.08t2 + 2.3t + 31

where L is the minimum normal height (in inches), H is the maximum normal height (in inches), and t is the age (in months).

c Source: Journal of Dairy Science

a. What is the normal height range for an 18-month-old Holstein heifer?

b. Describe the end behavior of eachfunction’s graph.

c. Graph the two height functions.

d. Writing Suppose a veterinarian examines a Holstein heifer that is 43 inches tall. About how old do you think the cow is? How did you get your answer?

EXAMINING END BEHAVIOR Use a spreadsheet or a graphing

calculator to evaluate the polynomial functions ƒ(x) = x 3 and

g(x) = x 3 º 2x 2 + 4x + 5 for the given values of x.

88. Copy and complete the table.

89. Use the results of Exercise 88 to complete this statement:

As x ˘ +‡, }g

ƒ

(

(

x

x

)

)} ˘

ooo

? .

Explain how this statement shows that the functions ƒ and g have the same end behavior as x ˘ +‡.

SIMPLIFYING EXPRESSIONS Simplify the expression. (Review 1.2 for 6.3)

90. x + 3 º 2x º x + 2 91. º2x2 + 3x + 4x + 2x2 92. º3x2 + 1 º (x2 + 2)

93. x2 + x + 1 + 3(x º 4) 94. 4x º 2x2 + 3 º x2 º 4 95. x2 º 1 º (2x2 + x º 3)

STANDARD FORM Write the quadratic function in standard form.

(Review 5.1 for 6.3)

96. y = º4(x º 2)2 + 5 97. y = º2(x + 6)(x º 5) 98. y = 2(x º 7)(x + 4)

99. y = 4(x º 3)2 º 24 100. y = º(x + 5)2 + 12 101. y = º3(x º 5)2 + 3

SOLVING QUADRATIC EQUATIONS Solve the equation. (Review 5.4)

102. x2 = º9 103. x2 = º5 104. º3x2 + 1 = 7

105. 4x2 + 15 = 3 106. 6x2 + 5 = 2x2 + 1 107. x2 = 7x2 + 1

108. x2 º 4 = º3x2 º 24 109. 3x2 + 5 = 5x2 + 10 110. 5x2 + 2 = º2x2 +1

MIXED REVIEW


TestPreparation

★★Challenge

A heifer is a young cow that has

not yet had calves.

x ƒ(x) g (x)

50 ? ? ?

100 ? ? ?

500 ? ? ?

1000 ? ? ?

5000 ? ? ?

ƒ(x)}g (x)

EXTRA CHALLENGE


N O P Q R S T U

6.2 Technology Activity 337

Setting a Good Viewing Window

When you graph a polynomial function with a graphing calculator, you must

choose a viewing window that displays the important characteristics of the

graph. Use what you know about end behavior to find such a viewing window.

c EXAMPLE

Graph ƒ(x) = 0.2x3 º 5x2 + 38x º 97.

c SOLUTION

Graph the function using the

standard viewing window.

Adjust the horizontal scale and the vertical scale until you see the graph’s

end behavior and any points where it turns. A good viewing window for this

graph is º10 ≤ x ≤ 20 and º20 ≤ y ≤ 10.

c EXERCISES

Find intervals for x and y that describe a good viewing window for the graph of

the polynomial function.

1. ƒ(x) = x3 + 6x2 º 11x + 3 2. ƒ(x) = ºx3 + 25x2 + 4

3. ƒ(x) = x4 º 5x2 + 6 4. ƒ(x) = ºx4 º 3x3 + x2 º x + 5

5. ƒ(x) = ºx5 + 5x3 º 4x + 10 6. ƒ(x) = x5 º 10x4 + 35x3 º 50x2 + 24x

7. EDUCATION For 1983 to 1996, the amount P (in millions of dollars) spentby public elementary and secondary schools and the amount R (in millions ofdollars) spent by private elementary and secondary schools can be modeled by

P = 11.7x4 º 340x3 + 2931x2 + 1560x + 182,000

R = 0.422x4 º 9.84x3 + 44.9x2 + 779x + 15,900

where x is the number of years since 1983. Find intervals for the horizontal and vertical axes that describe a good viewing window for the graphs of both functions. c Source: U.S. National Center for Education Statistics

2

1

Using Technology

Graphing Calculator Activity for use with Lesson 6.2ACTIVITY 6.2

STUDENT HELP

KEYSTROKE

HELP

See keystrokes for several models ofcalculators atwww.mcdougallittell.com

INT

ERNET

º10 ≤ x ≤ 10, º10 ≤ y ≤ 10

º10 ≤ x ≤ 20, º10 ≤ y ≤ 10 º10 ≤ x ≤ 20, º20 ≤ y ≤ 10

N O P Q R S T U


Adding, Subtracting, andMultiplying Polynomials

ADDING, SUBTRACTING, AND MULTIPLYING

To add or subtract polynomials, add or subtract the coefficients of like terms. You canuse a vertical or horizontal format.

Adding Polynomials Vertically and Horizontally

Add the polynomials.

a. 3x3 + 2x2 º x º 7

+ x3 º 10x2 º x + 8

4x3 º 8x2 º x + 1

b. (9x3 º 2x + 1) + (5x2 + 12x º 4) = 9x3 + 5x2 º 2x + 12x + 1 º 4

(9x3 º 2x + 1) + (5x2 + 12x º 4) = 9x3 + 5x2 + 10x º 3

Subtracting Polynomials Vertically and Horizontally

Subtract the polynomials.

a. º( 8x3 º 3x2 º 2x + 9 º 8x3 º 3x2 º 2x + 9º (2x

3 + 6x2 º x + 1) º2x

3 º 6x2 + x º 1 Add the opposite.

º 6x3 º 9x2 º x + 8

b. (2x2 + 3x) º (3x2 + x º 4) = 2x2 + 3x º 3x

2 º x + 4 Add the opposite.

(2x2 + 3x) º (3x2 + x º 4) = ºx2 + 2x + 4

. . . . . . . . . .

To multiply two polynomials, each term of the first polynomial must be multiplied byeach term of the second polynomial.

Multiplying Polynomials Vertically

Multiply the polynomials.

ºx3 ºx2 + 2x + 14ª ºx3 º 2x º 13

ºx3 3x2 º 6x º 12 Multiply ºx 2 + 2x + 4 by º3.

ºx3 + 2x2 + 4x Multiply ºx 2 + 2x + 4 by x.

ºx3 + 5x2 º 2x º 12 Combine like terms.

E X A M P L E 3

E X A M P L E 2

E X A M P L E 1

GOAL 1

Add, subtract, and

multiply polynomials.

Use polynomial

operations in real-lifeproblems, such as finding net

farm income in Example 7.

. To combine real-lifepolynomial models into a new

model, such as the model for

the power needed to keep a

bicycle moving at a certain

speed in Ex. 66.


GOAL 2

GOAL 1


6.3R

EAL LIFE

REA

L LIFE

STUDENT HELP

Look Back For help with simplifyingexpressions, see p. 251.

V W X Y Z [ \ ]

6.3 Adding, Subtracting, and Multiplying Polynomials 339

Multiplying Polynomials Horizontally


(x º 3)(3x2 º 2x º 4) = (x º 3)3x

2 º (x º 3)2x º (x º 3)4

(x º 3)(3x2 º 2x º 4) = 3x3 º 9x2 º 2x2 + 6x º 4x + 12

(x º 3)(3x2 º 2x º 4) = 3x3 º 11x2 + 2x + 12

Multiplying Three Binomials


(x º 1)(x + 4)(x + 3) = (x2 + 3x º 4)(x + 3)

= (x2 + 3x º 4)x + (x2 + 3x º 4)3

= x3 + 3x2 º 4x + 3x2 + 9x º 12

= x3 + 6x2 + 5x º 12

. . . . . . . . . .

Some binomial products occur so frequently that it is worth memorizing their special

product patterns. You can verify these products by multiplying.

Using Special Product Patterns


a. (4n º 5)(4n + 5) = (4n)2 º 52 Sum and difference

(4n º 5)(4n + 5) = 16n2 º 25

b. (9y º x2)2 = (9y)2 º 2(9y)(x2) + (x2)2 Square of a binomial

(9 º x2)2 = 81y2 º 18x2y + x4

c. (ab + 2)3 = (ab)3 + 3(ab)2(2) + 3(ab)(2)2 + 23 Cube of a binomial

(ab + 2)3 = a3b3 + 6a2b2 + 12ab + 8

E X A M P L E 6

E X A M P L E 5

E X A M P L E 4

SUM AND DIFFERENCE Example

(a + b)(a º b) = a2 º b2 (x + 3)(x º 3) = x2 º 9

SQUARE OF A BINOMIAL

(a + b)2 = a2 + 2ab + b2 (y + 4)2 = y2 + 8y + 16

(a º b)2 = a2 º 2ab + b2 (3t2 º 2)2 = 9t4 º 12t2 + 4

CUBE OF A BINOMIAL

(a + b)3 = a3 + 3a2b + 3ab2 + b3 (x + 1)3 = x 3 + 3x2 + 3x + 1

(a º b)3 = a3 º 3a2b + 3ab2 º b3 (p º 2)3 = p3 º 6p2 + 12p º 8

SPECIAL PRODUCT PATTERNS

STUDENT HELP

Look Back For help with multiplyingbinomials, see p. 251.

V W X Y Z [ \ ]


USING POLYNOMIAL OPERATIONS IN REAL LIFE

Subtracting Polynomial Models

FARMING From 1985 through 1995, the gross farm income G and farm expenses E(in billions of dollars) in the United States can be modeled by

G = º0.246t2 + 7.88t + 159 and E = 0.174t2 + 2.54t + 131

where t is the number of years since 1985. Write a model for the net farm income Nfor these years. c Source: U.S. Department of Agriculture

SOLUTION

To find a model for the net farm income, subtract the expenses model from the gross income model.

º0.246t2 + 7.88t + 159º (0.174t2 + 2.54t + 131)

º0.420t2 + 5.34t + 28

c The net farm income can be modeled by N = º0.42t2 + 5.34t + 28.

The graphs of the models are shown. Although Gand E both increase, the net income N eventuallydecreases because E increases faster than G.

Multiplying Polynomial Models

From 1982 through 1995, the number of softbound books N (in millions) sold in theUnited States and the average price per book P (in dollars) can be modeled by

N = 1.36t2 + 2.53t + 1076 and P = 0.314t + 3.42

where t is the number of years since 1982. Write a model for the total revenue Rreceived from the sales of softbound books. What was the total revenue fromsoftbound books in 1990? c Source: Book Industry Study Group, Inc.

SOLUTION

To find a model for R, multiply the models for N and P.

c The total revenue can be modeled by R = 0.427t3 + 5.45t2 + 347t + 3680. The graph of the revenue model is shown at the right. By substituting t = 8 into the model for R, you cancalculate that the revenue was about $7020 million, or $7.02 billion, in 1990.

1.36t2 + 2.53t + 1076ª 0.314t + 3.42}}}}}}}}}}}}}}

4.6512t2 + 8.6526t + 3679.920.42704t3 + 0.79442t2 + 337.864t

}}}}}}}}}}}}}}}}}}}}

0.42704t3 + 5.44562t2 + 346.5166t + 3679.92

E X A M P L E 8

E X A M P L E 7

GOAL 2

Years since 1985

2 4

Bil

lio

ns

of

do

lla

rs

150

250

50

60

0

200

100

8 10^

Farming

Years since 1982

2 4

Mil

lio

ns

of

do

lla

rs

60

12,000

8 10

10,000

8000

6000

4000

2000

012

_^

Softbound Books

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Publishing

FARMING Thenumber of farms in

the United States has beendecreasing steadily sincethe 1930s. However, theaverage size of farms hasbeen increasing.

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FOCUS ON

APPLICATIONS

` a b c d e f g


1. When you add or subtract polynomials, you add or subtract the coefficients of ooo

? .

2. ERROR ANALYSIS Describe the error in the subtraction shown below.

3. When you multiply a polynomial of degree 2 by a polynomial of degree 4, whatis the degree of the product?

Perform the indicated operation.

4. (4x2 + 3) + (3x2 + 8) 5. (2x3 º 4x2 + 5) + (ºx2 º 3x + 1)

6. (x2 + 7x º 5) º (3x2 + 1) 7. (x2 + 1) º (3x2 º 4x + 3)

8. (x + 2)(2x2 + 3) 9. (x2 + 3x + 10)(4x2 º 2x º 7)

10. (x º 1)(2x + 1)(x + 5) 11. (º3x + 1)3

12. Write a polynomial model in standard form for the volume of the rectangular prism shown at the right.

ADDING AND SUBTRACTING POLYNOMIALS Find the sum or difference.

13. (8x2 + 1) + (3x2 º 2) 14. (3x3 + 10x + 5) º (x3 º 4x + 6)

15. (x2 º 6x + 5) º (x2 + x º 2) 16. (16 º 13x) + (10x º 11)

17. (7x3 º 1) º (15x3 + 4x2 º x + 3) 18. 8x + (14x + 3 º 41x2 + x3)

19. (4x2 º 11x + 10) + (5x º 31) 20. (9x3 º 4 + x2 + 8x) º (7x3 º 3x + 7)

21. (º3x3 + x º 11) º (4x3 + x2 º x) 22. (6x2 º 19x + 5) º (19x2 º 4x + 9)

23. (10x3 º 4x2 + 3x) º (x3 º x2 + 1) 24. (50x º 3) + (8x3 + 7x2 + x + 4)

25. (10x º 3 + 7x2 ) + (x3 º 2x + 17) 26. (3x3 º 5x4 º 10x + 1) + (17x4 º x3)

MULTIPLYING POLYNOMIALS Find the product of the polynomials.

27. x(x2 + 6x º 7) 28. 10x2(x º 5) 29. º4x(x2 º 8x + 3)

30. 5x(3x2 º x + 3) 31. (x º 4)(x º 7) 32. (x + 9)(x º 2)

33. (x + 3)(x2 º 4x + 9) 34. (x + 8)(x2 º 7x º 3) 35. (2x + 5)(3x3 º x2 + x)

36. (6x + 2)(2x2 º 6x + 1) 37. (x + 11)(x2 º 5x + 9) 38. (4x2 º 1)(x2 º 6x + 9)

39. (x º 1)(x3 + 2x2 + 2) 40. (x + 1)(5x3 º x2 + x º 4)

41. (3x2 º 2)(x2 + 4x + 3) 42. (ºx3 º 2)(x2 + 3x º 3)

43. (x2 + x + 4)(2x2 º x + 1) 44. (x2 º x º 3)(x2 + 4x + 2)


GEOMETRY CONNECTION

GUIDED PRACTICE

Vocabulary Check ✓Concept Check ✓

Skill Check ✓

STUDENT HELP


STUDENT HELP

HOMEWORK HELP

Examples 1, 2: Exs. 13–26Examples 3, 4: Exs. 27–44Example 5: Exs. 45–52Example 6: Exs. 53–61Example 7: Exs. 64, 65, 69Example 8: Exs. 66–68

x 2 3

x 2 2x 1 3

(x2 º 3x + 4) º (x2 + 7x º 2) = x2 – 3x + 4 – x2 + 7x º 2

= 4x + 2

` a b c d e f g


MULTIPLYING THREE BINOMIALS Find the product of the binomials.

45. (x + 9)(x º 2)(x º 7) 46. (x + 3)(x º 4)(x º 5)

47. (x + 5)(x + 7)(ºx + 1) 48. (2x º 3)(x + 7)(x + 6)

49. (x º 9)(x º 2)(3x + 2) 50. (x º 1)(º2x º 5)(x º 8)

51. (2x + 1)(3x + 1)(x + 4) 52. (4x º 1)(2x º 1)(3x º 2)

SPECIAL PRODUCTS Find the product.

53. (x + 7)(x º 7) 54. (x + 4)2 55. (4x º 3)3

56. (10x + 3)(10x º 3) 57. (6 º x2)2 58. (2y + 5x)2

59. (3x + 7)3 60. (7y º x)2 61. (2x + 3y)3

Write the volume of the figure as a polynomial in

standard form.

62. V = πr 2h 63. V = lwh

64. MOTOR VEHICLE SALES For 1983 through 1996, the number of cars C

(in thousands) and the number of trucks and buses T (in thousands) sold thatwere manufactured in the United States can be modeled by

C = º1.63t4 + 49.5t3 º 476t2 + 1370t + 6705

T = º1.052t4 + 31.6t3 º 296t2 + 1097t + 2290

where t is the number of years since 1983. Find a model that represents the totalnumber of vehicles sold that were manufactured in the United States. How manyvehicles were sold in 1990?

65. For 1980 through 1996, the population P (inthousands) of the United States and the number of people S (in thousands) age 85 and over can be modeled by

P = º0.804t4 + 26.9t3 º 262t2 + 3010t + 227,000

S = 0.0206t4 º 0.670t3 + 6.42t2 + 213t + 7740

where t is the number of years since 1980. Find a model that represents thenumber of people in the United States under the age of 85. How many peoplewere under the age of 85 in 1995?

DATA UPDATE of U.S. Bureau of the Census data at www.mcdougallittell.com

66. BICYCLING The equation P = 0.00267sF gives the power P (inhorsepower) needed to keep a certain bicycle moving at speed s (in miles perhour), where F is the force of road and air resistance (in pounds). On levelground this force is given by F = 0.0116s2 + 0.789. Write a polynomial function(in terms of s only) for the power needed to keep the bicycle moving at speed son level ground. How much power does a cyclist need to exert to keep thebicycle moving at 10 miles per hour?

INT

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SOCIAL STUDIES CONNECTION

2x 1 3

x 1 1

x

x 2 2

x 1 3

GEOMETRY CONNECTION

GERONTOLOGIST

A gerontologist studies the biological,psychological, and socio-logical phenomena associat-ed with old age. As people’slife expectancies haveincreased, demand forgerontologists has grown.

CAREER LINK


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FOCUS ON

CAREERS

h i j k l m n o


67. EDUCATION For 1980 through 1995, the number of degrees D (inthousands) earned by people in the United States and the percent of degrees P earned by women can be modeled by

D = º0.096t4 + 3t3 º 27t2 + 91t + 1700

P = 0.43t + 49

where t is the number of years since 1980. Find a model that represents thenumber of degrees W (in thousands) earned by women from 1980 to 1995. Howmany degrees were earned by women in 1991? c Source: U.S. Bureau of the Census

68. PUBLISHING From 1985 through 1993, the number of hardback books N (inmillions) sold in the United States and the average price per book P (in dollars)can be modeled by

N = º0.27t 3 + 3.9t2 + 7.9t + 650

P = 0.67t + 9.4

where t is the number of years since 1985. Write a model that represents the totalrevenue R (in millions of dollars) received from the sales of hardback books.What was the revenue in 1991?

69. PERSONAL FINANCE Suppose two brothers each make three deposits inaccounts earning the same annual interest rate r (expressed as a decimal).

Mark’s account is worth 6000(1 + r)3 + 8000(1 + r)2 + 9000(1 + r) on January 1, 2000. Find the value of Tom’s account on January 1, 2000. Then findthe total value of the two accounts on January 1, 2000. Write the total value as apolynomial in standard form.

70. MULTIPLE CHOICE What is the sum of 2x4 + 5x3 º 8x2 º x + 10 and 8x4 º 4x3 + x2 º x + 2?

¡A 10x4 + x3 º 9x2 + 12 ¡B 10x4 + x3 º 9x2 º 2x + 12

¡C 10x4 + x3 º 7x2 º 2x + 12 ¡D 10x4 + 9x3 º 7x2 º 2x + 12

71. MULTIPLE CHOICE (3x º 8)3 = ooo

?

¡A 27x3 º 216x2 + 576x º 512 ¡B 27x3 º 216x2 + 576x + 512

¡C 27x3 º 72x2 + 576x º 512 ¡D 27x3 º 216x2 + 72x º 512

72. FINDING A PATTERN Look at the following polynomials and theirfactorizations.

x2 º 1 = (x º 1)(x + 1)x3 º 1 = (x º 1)(x2 + x + 1)x4 º 1 = (x º 1)(x3 + x2 + x + 1)

a. Factor x5 º 1 and x6 º 1. Check your answers by multiplying.

b. In general, how can xn º 1 be factored? Show that this factorization works bymultiplying the factors.

★★Challenge

TestPreparation

EXTRA CHALLENGE


HOMEWORK HELP

Visit our Web sitewww.mcdougallittell.comfor help with problemsolving in Ex. 69.

INT

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STUDENT HELP

h i j k l m n o


SOLVING QUADRATIC EQUATIONS Solve the equation. (Review 5.2 for 6.4)

73. 4x2 º 36 = 0 74. x2 + 3x º 40 = 0 75. x2 + 16x + 64 = 0

76. x2 º x º 56 = 0 77. 2x2 º 7x º 15 = 0 78. 6x2 + 10x º 4 = 0

WRITING QUADRATIC FUNCTIONS Write a quadratic function in standard

form whose graph passes through the given points. (Review 5.8)

79. (º4, 0), (2, 0), (1, 6) 80. (10, 0), (1, 0), (4, 3)

81. (º6, 0), (6, 0), (º3, º9) 82. (º3, 0), (5, 0), (º2, 7)

SIMPLIFYING ALGEBRAIC EXPRESSIONS Simplify the expression. Tell which

properties of exponents you used. (Review 6.1)

83. x5 • 84. 85. º5º2y0

86. (4xº3)4 • S}

x2

6

}D2

87. 88.

Evaluate the expression. (Lesson 6.1)

1. 70 • 5º3 2. S}

49

}Dº23. S}

3

52}D2

4. 32 • (32 • 24)º1 5. (82 • 8º3)2 • 82 6.

Simplify the expression. (Lesson 6.1)

7. (º5)º2y0 8. (3x3y6)º2 9. (x3yº5)(x2y)2

10. (x2yº3)(xy2) 11. S Dº312.

Graph the polynomial function. (Lesson 6.2)

13. ƒ(x) = x4 º 2 14. ƒ(x) = º2x5 + 3 15. ƒ(x) = 3x3 + 5x º 2

16. ƒ(x) = ºx3 + x2 º 2 17. ƒ(x) = x3 º 2x 18. ƒ(x) = ºx4 º 3x + 6

Perform the indicated operation. (Lesson 6.3)

19. (7x3 + 8x º 11) + (3x2 º x + 8) 20. (º2x2 + 4x) + (5x2 º x º 11)

21. (º5x2 + 12x º 9) º (º7x2 º 6x º 7) 22. (3x2 + 4x º 1) º (ºx3 + 2x + 5)

23. (x + 5)(4x2 º x º 1) 24. (x º 3)(x + 2)(2x + 5)

25. (x º 6)3 26. (2x2 + 3)2

27. ASTRONOMY Suppose NASA launches a spacecraft that can travel at aspeed of 25,000 miles per hour in space. How long would it take the spacecraftto reach Jupiter if Jupiter is about 495,000,000 miles away? Use scientificnotation to get your answer. (Lesson 6.1)

x6yº2

}

xº1y5

2x}

y2

(25 • 32)º1

}

2º2 • 32

QUIZ 1 Self-Test for Lessons 6.1–6.3

6x4y2

}

30x2yº1

3x5y8

}

6xyº3

x4y5

}

xy3

1}

x2

MIXED REVIEW

p q r s t u v wFactoring and SolvingPolynomial Equations

FACTORING POLYNOMIAL EXPRESSIONS

In Chapter 5 you learned how to factor the following types of quadratic expressions.

TYPE EXAMPLE

General trinomial 2x2 º 5x º 12 = (2x + 3)(x º 4)

Perfect square trinomial x2 + 10x + 25 = (x + 5)2

Difference of two squares 4x2 º 9 = (2x + 3)(2x º 3)

Common monomial factor 6x2 + 15x = 3x(2x + 5)

In this lesson you will learn how to factor other types of polynomials.

In the activity you may have discovered how to factor the difference of two cubes.This factorization and the factorization of the sum of two cubes are given below.

GOAL 1

6.4 Factoring and Solving Polynomial Equations 345

Factor polynomial

expressions.

Use factoring to

solve polynomial equations,

as applied in Ex. 87.

. To solve real-lifeproblems, such as finding

the dimensions of a block

discovered at an underwater

archeological site in

Example 5.


GOAL 2

GOAL 1


6.4

The Difference of Two Cubes

Use the diagram to answer the questions.

Explain why a3 º b3 = + + .

For each of solid I, solid II, and solid III,write an algebraic expression for the solid’s volume. Leave your expressionsin factored form.

Substitute your expressions from Step 2 into the equation from Step 1. Use the resulting equationto factor a3 º b3 completely.

3

2

Volume ofsolid III

Volume ofsolid II

Volume ofsolid I

1

DevelopingConcepts

ACTIVITY

SUM OF TWO CUBES Example

a3 + b3 = (a + b)(a2 º ab + b2) x

3 + 8 = (x + 2)(x2 º 2x + 4)

DIFFERENCE OF TWO CUBES

a3 º b3 = (a º b)(a2 + ab + b2) 8x

3 º 1 = (2x º 1)(4x2 + 2x + 1)

SPECIAL FACTORING PATTERNS

a

a

ab

bb

I

IIIII

REA

L LIFE

REA

L LIFE

p q r s t u v w


Factoring the Sum or Difference of Cubes

Factor each polynomial.

a. x3 + 27 b. 16u5 º 250u2

SOLUTION

a. x3 + 27 = x3 + 33 Sum of two cubes

= (x + 3)(x2 º 3x + 9)

b. 16u5 º 250u2 = 2u2(8u3 º 125) Factor common monomial.

= 2u2f(2u)3 º 53g Difference of two cubes

= 2u2(2u º 5)(4u2 + 10u + 25)

. . . . . . . . . .

For some polynomials, you can pairs of terms that have acommon monomial factor. The pattern for this is as follows.

ra + rb + sa + sb = r(a + b) + s(a + b)

= (r + s)(a + b)

Factoring by Grouping

Factor the polynomial x3 º 2x2 º 9x + 18.

SOLUTION

x3 º 2x2 º 9x + 18 = x2(x º 2) º 9(x º 2) Factor by grouping.

= (x2 º 9)(x º 2)

= (x + 3)(x º 3)(x º 2) Difference of squares

. . . . . . . . . .

An expression of the form au2 + bu + c where u is any expression in x is said to be in The factoring techniques you studied in Chapter 5 cansometimes be used to factor such expressions.

Factoring Polynomials in Quadratic Form

Factor each polynomial.

a. 81x4 º 16 b. 4x6 º 20x4 + 24x2

SOLUTION

a. 81x4 º 16 = (9x2)2 º 42 b. 4x6 º 20x4 + 24x2 = 4x2(x4 º 5x2 + 6)

= (9x2 + 4)(9x2 º 4) = 4x2(x2 º 2)(x2 º 3)

= (9x2 + 4)(3x + 2)(3x º 2)

E X A M P L E 3

quadratic form.

E X A M P L E 2

factor by grouping

E X A M P L E 1

x y z { | } ~ �


SOLVING POLYNOMIAL EQUATIONS BY FACTORING

In Chapter 5 you learned how to use the zero product property to solve factorablequadratic equations. You can extend this technique to solve some higher-degreepolynomial equations.

Solving a Polynomial Equation

Solve 2x5 + 24x = 14x3.

SOLUTION

2x5 + 24x = 14x3 Write original equation.

2x5 º 14x3 + 24x = 0 Rewrite in standard form.

2x(x4 º 7x2 + 12) = 0 Factor common monomial.

2x(x2 º 3)(x2 º 4) = 0 Factor trinomial.

2x(x2 º 3)(x + 2)(x º 2) = 0 Factor difference of squares.

x = 0, x = Ï3w, x = ºÏ3w, x = º2, or x = 2 Zero product property

c The solutions are 0, Ï3w, ºÏ3w, º2, and 2. Check these in the original equation.

Solving a Polynomial Equation in Real Life

ARCHEOLOGY In 1980 archeologists at the ruins of Caesara discovered a hugehydraulic concrete block with a volume of 330 cubic yards. The block’s dimensionsare x yards high by 13x º 11 yards long by 13x º 15 yards wide. What is the height?

SOLUTION

= • •

Volume = (cubic yards)

Height = (yards)

Length = (yards)

Width = (yards)

330 =

0 = 169x3 º 338x2 + 165x º 330 Write in standard form.

0 = 169x2(x º 2) + 165(x º 2) Factor by grouping.

0 = (169x2 + 165)(x º 2)

c The only real solution is x = 2, so 13x º 11 = 15 and 13x º 15 = 11. The blockis 2 yards high. The dimensions are 2 yards by 15 yards by 11 yards.

(13x º 15)(13x º 11)x

13x º 15

13x º 11

x

330

WidthLengthHeightVolume

E X A M P L E 5

E X A M P L E 4

GOAL 2

VERBAL

MODEL

ALGEBRAIC

MODEL

LABELS

STUDENT HELP

Study TipIn the solution ofExample 4, do not divide both sides of theequation by a variable ora variable expression.Doing so will result inthe loss of solutions.

ARCHEOLOGIST

Archeologistsexcavate, classify, and dateitems used by ancientpeople. They may specializein a particular geographicalregion and/or time period.

CAREER LINK


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FOCUS ON

CAREERS

x y z { | } ~ �


1. Give an example of a polynomial in quadratic form that contains an x3-term.

2. State which factoring method you would use to factor each of the following.

a. 6x3 º 2x2 + 9x º 3 b. 8x3 º 125 c. 16x4 º 9

3. ERROR ANALYSIS What is wrong with the solutionat the right?

4. a. Factor the polynomial x3 + 1 into the product of alinear binomial and a quadratic trinomial.

b. Show that you can’t factor the quadratic trinomialfrom part (a).

Factor the polynomial using any method.

5. x6 + 125 6. 4x3 + 16x2 + x + 4 7. x4 º 1

8. 2x3 º 3x2 º 10x + 15 9. 5x3 º 320 10. x4 + 7x2 + 10

Find the real-number solutions of the equation.

11. x3 º 27 = 0 12. 3x3 + 7x2 º 12x = 28 13. x3 + 2x2 º 9x = 18

14. 54x3 = º2 15. 9x4 º 12x2 + 4 = 0 16. 16x8 = 81

17. BUSINESS The revenue R (in thousands of dollars) for a small business can be modeled by

R = t3 º 8t2 + t + 82

where t is the number of years since 1990. In what year did the revenue reach $90,000?

MONOMIAL FACTORS Find the greatest common factor of the terms in the

polynomial.

18. 14x2 + 8x + 72 19. 3x4 º 12x3 20. 7x + 28x2 º 35x3

21. 24x4 º 6x 22. 39x5 + 13x3 º 78x2 23. 145x9 º 17

24. 6x6 º 3x4 º 9x2 25. 72x9 + 15x6 + 9x3 26. 6x4 º 18x3 + 15x2

MATCHING Match the polynomial with its factorization.

27. 3x2 + 11x + 6 A. 2x3(x + 2)(x º 2)(x2 + 4)

28. x3 º 4x2 + 4x º 16 B. 2x(x + 4)(x º 4)

29. 125x3 º 216 C. (3x + 2)(x + 3)

30. 2x7 º 32x3 D. (x2 + 4)(x º 4)

31. 2x5 + 4x4 º 4x3 º 8x2 E. 2x2(x2 º 2)(x + 2)

32. 2x3 º 32x F. (5x º 6)(25x2 + 30x + 36)


GUIDED PRACTICE


Skill Check ✓

STUDENT HELP


2x4 º 18x2 = 0

2x2(x2 º 9) = 0

x2 º 9 = 0

(x + 3)(x º 3) = 0

x = º3 or x = 3

Ex. 3

� � � � � � � �


SUM OR DIFFERENCE OF CUBES Factor the polynomial.

33. x3 º 8 34. x3 + 64 35. 216x3 + 1 36. 125x3 º 8

37. 1000x3 + 27 38. 27x3 + 216 39. 32x3 º 4 40. 2x3 + 54

GROUPING Factor the polynomial by grouping.

41. x3 + x2 + x + 1 42. 10x3 + 20x2 + x + 2 43. x3 + 3x2 + 10x + 30

44. x3 º 2x2 + 4x º 8 45. 2x3 º 5x2 + 18x º 45 46. º2x3 º 4x2 º 3x º 6

47. 3x3 º 6x2 + x º 2 48. 2x3 º x2 + 2x º 1 49. 3x3 º 2x2 º 9x + 6

QUADRATIC FORM Factor the polynomial.

50. 16x4 º 1 51. x4 + 3x2 + 2 52. x4 º 81

53. 81x4 º 256 54. 4x4 º 5x2 º 9 55. x4 + 10x2 + 16

56. 81 º 16x4 57. 32x6 º 2x2 58. 6x5 º 51x3 º 27x

CHOOSING A METHOD Factor using any method.

59. 18x3 º 2x2 + 27x º 3 60. 6x3 + 21x2 + 15x 61. 4x4 + 39x2 º 10

62. 8x3 º 12x2 º 2x + 3 63. 8x3 º 64 64. 3x4 º 300x2

65. 3x4 º 24x 66. 5x4 + 31x2 + 6 67. 3x4 + 9x3 + x2 + 3x

SOLVING EQUATIONS Find the real-number solutions of the equation.

68. x3 º 3x2 = 0 69. 2x3 º 6x2 = 0 70. 3x4 + 15x2 º 72 = 0

71. x3 + 27 = 0 72. x3 + 2x2 º x = 2 73. x4 + 7x3 º 8x º 56 = 0

74. 2x4 º 26x2 + 72 = 0 75. 3x7 º 243x3 = 0 76. x3 + 3x2 º 2x º 6 = 0

77. 8x3 º 1 = 0 78. x3 + 8x2 = º16x 79. x3 º 5x2 + 5x º 25 = 0

80. 3x4 + 3x3 = 6x2 + 6x 81. x4 + x3 º x = 1 82. 4x4 + 20x2 = º25

83. º2x6 = 16 84. 3x7 = 81x4 85. 2x5 º 12x3 = º16x

86. Writing You have now factored several different types of polynomials.Explain which factoring techniques or patterns are useful for factoring binomials,trinomials, and polynomials with more than three terms.

87. PACKAGING A candy factory needs a box that has a volume of 30 cubicinches. The width should be 2 inches less than the height and the length shouldbe 5 inches greater than the height. What should the dimensions of the box be?

88. MANUFACTURING A manufacturer wants tobuild a rectangular stainless steel tank with a holdingcapacity of 500 gallons, or about 66.85 cubic feet. Ifsteel that is one half inch thick is used for the walls ofthe tank, then about 5.15 cubic feet of steel is needed.The manufacturer wants the outside dimensions of the tank to be related as follows:

• The width should be one foot less than the length.

• The height should be nine feet more thanthe length.

What should the outside dimensions of the tank be?

x 2 1x

x 1 9

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 18–40,59–67

Example 2: Exs. 18–32,41–49, 59–67

Example 3: Exs. 18–32,50–67

Example 4: Exs. 68–85Example 5: Exs. 87–92

HOMEWORK HELP


INT

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STUDENT HELP

� � � � � � � �


89. CITY PARK For the city parkcommission, you are designing a marbleplanter in which to plant flowers. You wantthe length of the planter to be six times theheight and the width to be three times theheight. The sides should be one foot thick.Since the planter will be on the sidewalk, itdoes not need a bottom. What should theouter dimensions of the planter be if it is tohold 4 cubic feet of dirt?

SCULPTURE In Exercises 90 and 91,

refer to the sculpture shown in the picture.

90. The “cube” portion of the sculpture is actually a rectangular prism with dimensions x feet by 5x º 10 feet by 2x º 1 feet. The volume of the prism is 25 cubic feet. What are the dimensions of the prism?

91. Suppose a pyramid like the one in the sculpture is 3x feet high and has a squarebase measuring x º 5 feet on each side. If the volume is 250 cubic feet, what

are the dimensions of the pyramid? (Use the formula V = }13

}Bh.)

92. CRAFTS Suppose you have 250 cubic inches of clay with which to make arectangular prism for a sculpture. If you want the height and width each to be5 inches less than the length, what should the dimensions of the prism be?

93. MULTIPLE CHOICE The expression (3x º 4)(9x2 + 12x + 16) is thefactorization of which of the following?

¡A 27x3 º 8 ¡B 27x3 + 36x2¡C 27x3 º 64 ¡D 27x3 + 64

94. MULTIPLE CHOICE Which of the following is the factorization of x3 º 8?

¡A (x º 2)(x2 + 4x + 4) ¡B (x + 2)(x2 º 2x + 4)

¡C (x + 2)(x2 º 4x + 4) ¡D (x º 2)(x2 + 2x + 4)

95. MULTIPLE CHOICE What are the real solutions of the equation x5 = 81x?

¡A x = ±3, ±3i ¡B x = 0, ±9

¡C x = 0, ±3, ±3i ¡D x = 0, ±3

96. Explain how thefigure shown at the right can be used as ageometric factoring model for the sum of two cubes.

a3 + b3 = (a + b)(a2 º ab + b2)

Factor the polynomial.

97. 30x2y + 36x2 º 20xy º 24x

98. 2x7 º 127x

GEOMETRY CONNECTION

6x – 2

3x – 2

6x

3x

x

a

a

a

bb

b

TestPreparation

★★Challenge

“Charred Sphere, Cube, and Pyramid”

by David Nash

Ex. 96

EXTRA CHALLENGE


� � � � � � � �


SIMPLIFYING EXPRESSIONS Simplify the expression. (Review 6.1 for 6.5)

99. }36

6

x

x3

3

y

yº

9

2} 100. 101. }

4

7

9

2

x

xº

º

3

3

y

yº

2

2}

SYNTHETIC SUBSTITUTION Use synthetic substitution to evaluate the

polynomial function for the given value of x. (Review 6.2 for 6.5)

102. ƒ(x) = 3x4 + 2x3 º x2 º 12x + 1, x = 3

103. ƒ(x) = 2x5 º x3 + 7x + 1, x = 3

104. SEWING At the fabric store you are buying solid fabric at $4 per yard, printfabric at $6 per yard, and a pattern for $8. Write an equation for the amount youspend as a function of the amount of solid and print fabric you buy. (Review 3.5)

5º2x2yº1

}}

52xy3

MIXED REVIEW

Solving Polynomial Equations

THENTHEN

APPLICATION LINK


INT

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1545

IN 2000 B.C. the Babylonians solved polynomial equations by referring totables of values. One such table gave the values of y3 + y2. To be able touse this table, the Babylonians sometimes had to manipulate theequation, as shown below.

ax3 + bx2 = c Write original equation.

}a

b

3x3

3

} + }a

b

2x2

2

} = }a

b

2

3

c} Multiply by }

a

b

2

3} .

S}a

b

x}D3

+ S}a

b

x}D2

= }a

b

2

3

c} Re-express cubes and squares.

Then they would find in the y3 + y2 column of the table.

Because they knew that the corresponding y-value was equal to

}abx}, they could conclude that x = }

b

a

y}.

1. Calculate y3 + y2 for y = 1, 2, 3, . . . , 10. Record the values in a table.

Use your table and the method discussed above to solve the equation.

2. x3 + x2 = 252 3. x3 + 2x2 = 288 4. 3x3 + x2 = 90

5. 2x3 + 5x2 = 2500 6. 7x3 + 6x2 = 1728 7. 10x3 + 3x2 = 297

TODAY computers use polynomial equations to accomplish many things, such as making robots move.

a2c}

b3

NOWNOW

Chinese solve

cubic equations.

Cardano solves

cubic equations.Babylonians use tables. 1994

Polynomials are

used to program

NASA robot.

2000 B .C .

A .D . 1100

� � � � � � � �The Remainder and FactorTheorems

DIVIDING POLYNOMIALS

When you divide a polynomial ƒ(x) by a divisor d(x), you get a quotient polynomial

q(x) and a remainder polynomial r(x). We write this as }ƒ

d

(

(

x

x

)

)} = q(x) + }

d

r(

(

x

x

)

)}. The

degree of the remainder must be less than the degree of the divisor.

Example 1 shows how to divide polynomials using a method called

Using Polynomial Long Division

Divide 2x4 + 3x3 + 5x º 1 by x2 º 2x + 2.

SOLUTION

Write division in the same format you would use when dividing numbers. Include a“0” as the coefficient of x2.

}

2

x

x2

4

} }

7

x

x2

3

} }

1

x

02

x2

}

2x2 + 17x + 10

x2 º 2x + 2 q2wx4w +w 3wx3w +w 1w0wx2w +w 1w5wxwºw 1w1w

2x4 º 4x3 + 14x2 Subtract 2x 2(x 2 º 2x + 2).

7x3 º 14x2 + 15x

7x3 º 14x2 + 14x Subtract 7x (x 2 º 2x + 2).

10x2 º 19x º 11

10x2 º 20x + 20 Subtract 10(x 2 º 2x + 2).

11x º 21 remainder

Write the result as follows.

c = 2x2 + 7x + 10 +

✓CHECK You can check the result of a division problem by multiplying the divisor

by the quotient and adding the remainder. The result should be the dividend.

(2x2 + 7x + 10)(x2 º 2x + 2) + 11x º 21

= 2x2(x2 º 2x + 2) + 7x(x2 º 2x + 2) + 10(x2 º 2x + 2) + 11x º 21

= 2x4 º 4x3 + 4x2 + 7x3 º 14x2 + 14x + 10x2 º 20x + 20 + 11x º 21

= 2x4 + 3x3 + 5x º 1 ✓

11x º 21}}

x2 º 2x + 2

2x4 + 3x3 + 5x º 1}}}

x2 º 2x + 2

E X A M P L E 1

long division.polynomial

GOAL 1


Divide polynomials

and relate the result to the

remainder theorem and the

factor theorem.

Use polynomial

division in real-life problems,

such as finding a production

level that yields a certain

profit in Example 5.

. To combine two real-lifemodels into one new model,

such as a model for money

spent at the movies each

year in Ex. 62.


GOAL 2

GOAL 1


6.5R

EAL LIFE

REA

L LIFE

At each stage, divide the term withthe highest power in what’s left ofthe dividend by the first term of thedivisor. This gives the next term ofthe quotient.

� � � � � � � �

6.5 The Remainder and Factor Theorems 353

In the activity you may have discovered that ƒ(2) gives you the remainder when ƒ(x)is divided by x º 2. This result is generalized in the remainder theorem.

You may also have discovered in the activity that synthetic substitution gives thecoefficients of the quotient. For this reason, synthetic substitution is sometimes called

It can be used to divide a polynomial by an expression of theform x º k.

Using Synthetic Division

Divide x3 + 2x2 º 6x º 9 by (a) x º 2 and (b) x + 3.

SOLUTION

a. Use synthetic division for k = 2.

c = x2 + 4x + 2 + }xºº

52

}

b. To find the value of k, rewrite the divisor in the form x º k. Because x + 3 = x º (º3), k = º3.

c = x2 º x º 3x3 + 2x2 º 6x º 9}}

x + 3

x3 + 2x2 º 6x º 9}}

x º 2

E X A M P L E 2

synthetic division.

If a polynomial ƒ(x) is divided by x º k, then the remainder is r = ƒ(k).

REMAINDER THEOREM

STUDENT HELP

Study TipNotice that syntheticdivision could not havebeen used to divide thepolynomials in Example 1because the divisor,x

2 º 2x + 2, is not of theform x º k.

2 1 2 º6 º92 8 4

1 4 2 º5

º3 1 2 º6 º9º3 3 9

1 º1 º3 0

Investigating Polynomial Division

Let ƒ(x) = 3x3 º 2x2 + 2x º 5.

Use long division to divide ƒ(x) by x º 2. What is the quotient? What is theremainder?

Use synthetic substitution to evaluate ƒ(2). How is ƒ(2) related to theremainder? What do you notice about the other constants in the last row ofthe synthetic substitution?

2

1

DevelopingConcepts

ACTIVITY

� � � � � � � �


In part (b) of Example 2, the remainder is 0. Therefore, you can rewrite the result as:

x3 + 2x2 º 6x º 9 = (x2 º x º 3)(x + 3)

This shows that x + 3 is a factor of the original dividend.

Recall from Chapter 5 that the number k is called a zero of the function ƒ becauseƒ(k) = 0.

Factoring a Polynomial

Factor ƒ(x) = 2x3 + 11x2 + 18x + 9 given that ƒ(º3) = 0.

SOLUTION

Because ƒ(º3) = 0, you know that x º (º3) or x + 3 is a factor of ƒ(x). Use synthetic division to find the other factors.

The result gives the coefficients of the quotient.

2x3 + 11x2 + 18x + 9 = (x + 3)(2x2 + 5x + 3)

= (x + 3)(2x + 3)(x + 1)

Finding Zeros of a Polynomial Function

One zero of ƒ(x) = x3 º 2x2 º 9x + 18 is x = 2. Find the other zeros of the function.

SOLUTION

To find the zeros of the function, factor ƒ(x) completely. Because ƒ(2) = 0, you knowthat x º 2 is a factor of ƒ(x). Use synthetic division to find the other factors.

The result gives the coefficients of the quotient.

ƒ(x) = (x º 2)(x2 º 9) Write ƒ(x) as a product of two factors.

= (x º 2)(x + 3)(x º 3) Factor difference of squares.

c By the factor theorem, the zeros of ƒ are 2, º3, and 3.

E X A M P L E 4

E X A M P L E 3

A polynomial ƒ(x) has a factor x º k if and only if ƒ(k) = 0.

FACTOR THEOREM

º3 2 11 18 9º6 º15 º9

2 5 3 0

2 1 º2 º9 182 0 º18

1 0 º9 0

HOMEWORK HELP


INT

ERNET

STUDENT HELP

� � � � � � � �


USING POLYNOMIAL DIVISION IN REAL LIFE

In business and economics, a function that gives the price per unit p of an item interms of the number x of units sold is called a demand function.

Using Polynomial Models

ACCOUNTING You are an accountant for a manufacturer of radios. The demandfunction for the radios is p = 40 º 4x2 where x is the number of radios produced inmillions. It costs the company $15 to make a radio.

a. Write an equation giving profit as a function of the number of radios produced.

b. The company currently produces 1.5 million radios and makes a profit of$24,000,000, but you would like to scale back production. What lesser number ofradios could the company produce to yield the same profit?

SOLUTION

a.

b. Substitute 24 for P in the function you wrote in part (a).

24 = º4x3 + 25x

0 = º4x3 + 25x º 24

You know that x = 1.5 is one solution of the equation. This implies that x º 1.5is a factor. So divide to obtain the following:

º2(x º 1.5)(2x2 + 3x º 8) = 0

Use the quadratic formula to find thatx ≈ 1.39 is the other positive solution.

c The company can make the sameprofit by selling 1,390,000 units.

✓CHECK Graph the profit function to

confirm that there are two production

levels that produce a profit of

$24,000,000.

E X A M P L E 5

GOAL 2

Profit = Revenue º Cost

= • º •

Profit = (millions of dollars)

Price per unit = (dollars per unit)

Number of units = (millions of units)

Cost per unit = 15 (dollars per unit)

= º 15

P = º4x3 + 25x

xx(40 º 4x2)P

x

40 º 4x2

P

Numberof units

Costper unit

Numberof units

Priceper unitProfit

LABELS

VERBAL

MODEL

ALGEBRAIC

MODEL

Number of units (millions)

1.2

Pro

fit

(mil

lio

ns

of

do

lla

rs)

24.0

23.5

23.0 ��

0 1.4 1.60

Radio Production

PROBLEMSOLVING

STRATEGY

ACCOUNTANT

Most people think ofaccountants as working formany clients. However, it iscommon for an accountantto work for a single client,such as a company or thegovernment.

CAREER LINK


INT

ERNET

RE

AL LIFE

RE

AL LIFE

FOCUS ON

CAREERS

� � � � � � ¡


1. State the remainder theorem.

2. Write a polynomial division problem that you would use long division to solve.Then write a polynomial division problem that you would use synthetic divisionto solve.

3. Write the polynomial divisor, dividend, and quotient represented by thesynthetic division shown at the right.

Divide using polynomial long division.

4. (2x3 º 7x2 º 17x º 3) ÷ (2x + 3) 5. (x3 + 5x2 º 2) ÷ (x + 4)

6. (º3x3 + 4x º 1) ÷ (x º 1) 7. (ºx3 + 2x2 º 2x + 3) ÷ (x2 º 1)

Divide using synthetic division.

8. (x3 º 8x + 3) ÷ (x + 3) 9. (x4 º 16x2 + x + 4) ÷ (x + 4)

10. (x2 + 2x + 15) ÷ (x º 3) 11. (x2 + 7x º 2) ÷ (x º 2)

Given one zero of the polynomial function, find the other zeros.

12. ƒ(x) = x3 º 8x2 + 4x + 48; 4 13. ƒ(x) = 2x3 º 14x2 º 56x º 40; 10

14. BUSINESS Look back at Example 5. If the company produces 1 millionradios, it will make a profit of $21,000,000. Find another number of radios thatthe company could produce to make the same profit.

USING LONG DIVISION Divide using polynomial long division.

15. (x2 + 7x º 5) ÷ (x º 2) 16. (3x2 + 11x + 1) ÷ (x º 3)

17. (2x2 + 3x º 1) ÷ (x + 4) 18. (x2 º 6x + 4) ÷ (x + 1)

19. (x2 + 5x º 3) ÷ (x º 10) 20. (x3 º 3x2 + x º 8) ÷ (x º 1)

21. (2x4 + 7) ÷ (x2 º 1) 22. (x3 + 8x2 º 3x + 16) ÷ (x2 + 5)

23. (6x2 + x º 7) ÷ (2x + 3) 24. (10x3 + 27x2 + 14x + 5) ÷ (x2 + 2x)

25. (5x4 + 14x3 + 9x) ÷ (x2 + 3x) 26. (2x4 + 2x3 º 10x º 9) ÷ (x3 + x2 º 5)

USING SYNTHETIC DIVISION Divide using synthetic division.

27. (x3 º 7x º 6) ÷ (x º 2) 28. (x3 º 14x + 8) ÷ (x + 4)

29. (4x2 + 5x º 4) ÷ (x + 1) 30. (x2 º 4x + 3) ÷ (x º 2)

31. (2x2 + 7x + 8) ÷ (x º 2) 32. (3x2 º 10x) ÷ (x º 6)

33. (x2 + 10) ÷ (x + 4) 34. (x2 + 3) ÷ (x + 3)

35. (10x4 + 5x3 + 4x2 º 9) ÷ (x + 1) 36. (x4 º 6x3 º 40x + 33) ÷ (x º 7)

37. (2x4 º 6x3 + x2 º 3x º 3) ÷ (x º 3) 38. (4x4 + 5x3 + 2x2 º 1) ÷ (x + 1)


GUIDED PRACTICE


Skill Check ✓

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 15–26Example 2: Exs. 27–38Example 3: Exs. 39–46Example 4: Exs. 47–54Example 5: Exs. 60–62

STUDENT HELP

Extra Practiceto help you masterskills is on p. 948.

º3 1 º2 º9 18º3 15 º18

1 º5 6 0

� � � � � � ¡


FACTORING Factor the polynomial given that ƒ(k) = 0.

39. ƒ(x) = x3 º 5x2 º 2x + 24; k = º2 40. ƒ(x) = x3 º 3x2 º 16x º 12; k = 6

41. ƒ(x) = x3 º 12x2 + 12x + 80; k = 10 42. ƒ(x) = x3 º 18x2 + 95x º 126; k = 9

43. ƒ(x) = x3 º x2 º 21x + 45; k = º5 44. ƒ(x) = x3 º 11x2 + 14x + 80; k = 8

45. ƒ(x) = 4x3 º 4x2 º 9x + 9; k = 1 46. ƒ(x) = 2x3 + 7x2 º 33x º 18; k = º6

FINDING ZEROS Given one zero of the polynomial function, find the other zeros.

47. ƒ(x) = 9x3 + 10x2 º 17x º 2; º2 48. ƒ(x) = x3 + 11x2 º 150x º 1512; º14

49. ƒ(x) = 2x3 + 3x2 º 39x º 20; 4 50. ƒ(x) = 15x3 º 119x2 º 10x + 16; 8

51. ƒ(x) = x3 º 14x2 + 47x º 18; 9 52. ƒ(x) = 4x3 + 9x2 º 52x + 15; º5

53. ƒ(x) = x3 + x2 + 2x + 24; º3 54. ƒ(x) = 5x3 º 27x2 º 17x º 6; 6

You are given an expression for the volume of the

rectangular prism. Find an expression for the missing dimension.

55. V = 3x3 + 8x2 º 45x º 50 56. V = 2x3 + 17x2 + 40x + 25

POINTS OF INTERSECTION Find all points of intersection of the two graphs

given that one intersection occurs at x = 1.

57. 58.

59. LOGICAL REASONING You divide two polynomials and obtain the result

5x2 º 13x + 47 º }

x1+02

2}. What is the dividend? How did you find it?

60. COMPANY PROFIT The demand function for a type of camera is given by the model p = 100 º 8x2 where p is measured in dollars per camera and x ismeasured in millions of cameras. The production cost is $25 per camera. Theproduction of 2.5 million cameras yielded a profit of $62.5 million. What othernumber of cameras could the company sell to make the same profit?

61. FUEL CONSUMPTION From 1980 to 1991, the total fuel consumption T(in billions of gallons) by cars in the United States and the average fuelconsumption A (in gallons per car) can be modeled by

T = º0.026x3 + 0.47x2 º 2.2x + 72 and A = º8.4x + 580

where x is the number of years since 1980. Find a function for the number of carsfrom 1980 to 1991. About how many cars were there in 1990?

¢ £¤ ¥¦

y 5 x 3 2 6x 2

1 6x 1 3

y 5 2x 2 1 7x 2 2

§¨ ¥

¦y 5 x 3

1 x 2 2 5x

y 5 2x 2 2 4x 1 2

x 1 5

x 1 1

?

x 1 5

x 1 1

?

GEOMETRY CONNECTION

ALTERNATIVE

FUEL

Joshua and Kaia Tickell builtthe Green Grease Machine,which converts usedrestaurant vegetable oil intobiodiesel fuel. The Tickellsuse the fuel in their motorhome, the Veggie Van, as an alternative to the fuelreferred to in Ex. 61.

APPLICATION LINK


INT

ERNET

RE

AL LIFE

RE

AL LIFE

FOCUS ON

APPLICATIONS

© ª « ¬ ® ¯ °


62. MOVIES The amount M (in millions of dollars) spent at movie theaters from1989 to 1996 can be modeled by

M = º3.05x3 + 70.2x2 º 225x + 5070

where x is the number of years since 1989. The United States population P

(in millions) from 1989 to 1996 can be modeled by the following function:

P = 2.61x + 247

Find a function for the average annual amount spent per person at movie theatersfrom 1989 to 1996. On average, about how much did each person spend at movietheaters in 1989? c Source: Statistical Abstract of the United States

63. MULTIPLE CHOICE What is the result of dividing x3 º 9x + 5 by x º 3?

¡A x2 + 3x + 5 ¡B x2 + 3x ¡C x2 + 3x + }x º

53

}

¡D x2 + 3x º }x º

53

} ¡E x2 + 3x º 18 + }x

5º9

3}

64. MULTIPLE CHOICE Which of the following is a factor of the polynomial2x3 º 19x2 º 20x + 100?

¡A x + 10 ¡B x + 2 ¡C 2x º 5 ¡D x º 5 ¡E 2x + 5

65. COMPARING METHODS Divide the polynomial 12x3 º 8x2 + 5x + 2 by2x + 1, 3x + 1, and 4x + 1 using long division. Then divide the same

polynomial by x + }

12

}, x + }

13

}, and x + }

14

} using synthetic division. What do you

notice about the remainders and the coefficients of the quotients from the twotypes of division?

CHECKING SOLUTIONS Check whether the given ordered pairs are solutions

of the inequality. (Review 2.6)

66. x + 7y ≤ º8; (6, º2), (º2, º3) 67. 2x + 5y ≥ 1; (º2, 4), (8, º3)

68. 9x º 4y > 7; (º1, º4), (2, 2) 69. º3x º 2y < º6; (2, 0), (1, 4)

QUADRATIC FORMULA Use the quadratic formula to solve the equation.

(Review 5.6 for 6.6)

70. x2 º 5x + 3 = 0 71. x2 º 8x + 3 = 0 72. x2 º 10x + 15 = 0

73. 4x2 º 7x + 1 = 0 74. º6x2 º 9x + 2 = 0 75. 5x2 + x º 2 = 0

76. 2x2 + 3x + 5 = 0 77. º5x2 º x º 8 = 0 78. 3x2 + 3x + 1 = 0

POLYNOMIAL OPERATIONS Perform the indicated operation. (Review 6.3)

79. (x2 º 3x + 8) º (x2 + x º 1) 80. (14x2 º 15x + 3) + (11x º 7)

81. (8x3 º 1) º (22x3 + 2x2 º x º 5) 82. (x + 5)(x2 º x + 5)

83. CATERING You are helping your sister plan her wedding reception. The guests have chosen whether they would like the chicken dish or thevegetarian dish. The caterer charges $24 per chicken dish and $21 per vegetariandish. After ordering the dinners for the 120 guests, the caterer’s bill comes to$2766. How many guests requested chicken? (Lesson 3.2)

MIXED REVIEW

TestPreparation

★★Challenge

EXTRA CHALLENGE


© ª « ¬ ® ¯ °

6.6 Finding Rational Zeros 359

Finding Rational Zeros

USING THE RATIONAL ZERO THEOREM

The polynomial function

ƒ(x) = 64x3 + 120x2 º 34x º 105

has º}32

}, º}54

}, and }78

} as its zeros. Notice that the numerators of these zeros (º3, º5,

and 7) are factors of the constant term, º105. Also notice that the denominators (2, 4,and 8) are factors of the leading coefficient, 64. These observations are generalizedby the rational zero theorem.

Using the Rational Zero Theorem

Find the rational zeros of ƒ(x) = x3 + 2x2 º 11x º 12.

SOLUTION

List the possible rational zeros. The leading coefficient is 1 and the constant term is º12. So, the possible rational zeros are:

x = ±}11

}, ±}21

}, ±}31

}, ±}41

}, ±}61

}, ±}112}

Test these zeros using synthetic division.

Test x = 1: Test x = º1:

Since º1 is a zero of ƒ, you can write the following:

ƒ(x) = (x + 1)(x2 + x º 12)

Factor the trinomial and use the factor theorem.

ƒ(x) = (x + 1)(x2 + x º 12) = (x + 1)(x º 3)(x + 4)

c The zeros of ƒ are º1, 3, and º4.

E X A M P L E 1

GOAL 1

Find the rational

zeros of a polynomial

function.

Use polynomial

equations to solve real-lifeproblems, such as finding the

dimensions of a monument

in Ex. 60.

. To model real-lifequantities, such as the

volume of a representation

of the Louvre pyramid in

Example 3.


GOAL 2

GOAL 1


6.6

If ƒ(x) = anxn + . . . + a1x + a0 has integer coefficients, then every rational zero

of ƒ has the following form:

}p

q} =

factor of constant term a0}}}}factor of leading coefficient an

THE RATIONAL ZERO THEOREM

1 2 º11 º121 3 º8

1 3 º8 º20

1 2 º11 º12º1 º1 12

1 1 º12 0

1 º1

REA

L LIFE

REA

L LIFE

± ² ³ ´ µ ¶ · ¸


In Example 1, the leading coefficient is 1. When the leading coefficient is not 1, thelist of possible rational zeros can increase dramatically. In such cases the search canbe shortened by sketching the function’s graph—either by hand or by using agraphing calculator.

Using the Rational Zero Theorem

Find all real zeros of ƒ(x) = 10x4 º 3x3 º 29x2 + 5x + 12.

SOLUTION

List the possible rational zeros of ƒ: ±}11

}, ±}21

}, ±}31

}, ±}41

}, ±}61

},

±}112}, ±}

32

}, ±}15

}, ±}25

}, ±}35

}, ±}65

}, ±}152}, ±}

110}, ±}

130}, ±}

11

20}.

Choose values to check.

With so many possibilities, it is worth your time to sketchthe graph of the function. From the graph, it appears that

some reasonable choices are x = º}32

}, x = º}35

}, x = }45

},

and x = }32

}.

Check the chosen values using synthetic division.

º}32} is a zero.

Factor out a binomial using the result of the synthetic division.

ƒ(x) = Sx + }32

}D(10x3 º 18x2 º 2x + 8) Rewrite as a product of two factors.

= Sx + }32

}D(2)(5x3 º 9x2 º x + 4) Factor 2 out of the second factor.

= (2x + 3)(5x3 º 9x2 º x + 4) Multiply the first factor by 2.

Repeat the steps above for g(x) = 5x3 º 9x2 º x + 4.

Any zero of g will also be a zero of ƒ. The possible rational zeros of g are

x = ±1, ±2, ±4, ±}15

}, ±}25

}, and ±}45

}. The graph of ƒ shows that }45

} may be a zero.

}45} is a zero.

So ƒ(x) = (2x + 3)Sx º }45}D(5x2 º 5x º 5) = (2x + 3)(5x º 4)(x2 º x º 1).

Find the remaining zeros of ƒ by using the quadratic formula to solvex2 º x º 1 = 0.

c The real zeros of ƒ are º}32

}, }45

}, , and .1 º Ï5w}

21 + Ï5w}

2

E X A M P L E 2

}45

} 5 º9 º1 44 º4 º4

5 º5 º5 0

º}32

} 10 º3 º29 5 12º15 27 3 º12

10 º18 º2 8 0

± ² ³ ´ µ ¶ · ¸


SOLVING POLYNOMIAL EQUATIONS IN REAL LIFE

Writing and Using a Polynomial Model

You are designing a candle-making kit. Each kit will contain 25 cubic inches of candle waxand a mold for making a model of the pyramid-shaped building at the Louvre Museum in Paris, France. You want the height of the candle to be 2 inches less than the length ofeach side of the candle’s square base. Whatshould the dimensions of your candle mold be?

SOLUTION

The volume is V = }13

}Bh where B is the area of the base and h is the height.

The possible rational solutions are x = ±}11

}, ±}31

}, ±}51

}, ±}115}, ±}

215}, ±}

715}.

Use the possible solutions. Note that in this case, it makes sense to test only positivex-values.

So x = 5 is a solution. The other two solutions, which satisfy x2 + 3x + 15 = 0, are

x = and can be discarded because they are imaginary.

c The base of the candle mold should be 5 inches by 5 inches. The height of themold should be 5 º 2 = 3 inches.

º3 ± iÏ5w1w}}

2

E X A M P L E 3

GOAL 2

RE

AL LIFE

RE

AL LIFE

Crafts

PROBLEMSOLVING

STRATEGY= }

13

} • •

Volume = 25 (cubic inches)

Side of square base = x (inches)

Area of base = (square inches)

Height = (inches)

25 = }13

} Write algebraic model.

75 = x3 º 2x2 Multiply each side by 3 and simplify.

0 = x3 º 2x2 º 75 Subtract 75 from each side.

(x º 2)x2

x º 2

x2

HeightArea of

baseVolume

LABELS

VERBAL

MODEL

ALGEBRAIC

MODEL

1 1 º2 0 º751 º1 º1

1 º1 º1 º76

5 1 º2 0 º755 15 75

1 3 15 0

3 1 º2 0 º753 3 9

1 1 3 º66

5 is a solution.

I.M. PEI designedthe pyramid at the

Louvre. His geometricarchitecture can be seen inBoston, New York, Dallas,Los Angeles, Taiwan, Beijing,and Singapore.

RE

AL LIFE

RE

AL LIFE

FOCUS ON

PEOPLE

x

x

x 2 2

¹ º » ¼ ½ ¾ ¿ À


1. Complete this statement of the rational zero theorem: If a polynomial functionhas integer coefficients, then every rational zero of the function has the

form }pq} , where p is a factor of the

ooo

? and q is a factor of the ooo

? .

2. For each polynomial function, decide whether you can use the rational zerotheorem to find its zeros. Explain why or why not.

a. ƒ(x) = 6x2 º 8x + 4 b. ƒ(x) = 0.3x2 + 2x + 4.5 c. ƒ(x) = }14

}x2 º x + }87

}

3. Describe a method you can use to shorten the list of possible rational zeros whenusing the rational zero theorem.

List the possible rational zeros of ƒ using the rational zero theorem.

4. ƒ(x) = x3 + 14x2 + 41x º 56 5. ƒ(x) = x3 º 17x2 + 54x + 72

6. ƒ(x) = 2x3 + 7x2 º 7x + 30 7. ƒ(x) = 5x4 + 12x3 º 16x2 + 10

Find all the real zeros of the function.

8. ƒ(x) = x3 º 3x2 º 6x + 8 9. ƒ(x) = x3 + 4x2 º x º 4

10. ƒ(x) = 2x3 º 5x2 º 2x + 5 11. ƒ(x) = 2x3 º x2 º 15x + 18

12. ƒ(x) = x3 + 4x2 + x º 6 13. ƒ(x) = x3 + 5x2 º x º 5

14. CRAFTS Suppose you have 18 cubic inches of wax and you want to make acandle in the shape of a pyramid with a square base. If you want the height of thecandle to be 3 inches greater than the length of each side of the base, what shouldthe dimensions of the candle be?

LISTING RATIONAL ZEROS List the possible rational zeros of ƒ using the

rational zero theorem.

15. ƒ(x) = x4 + 2x2 º 24 16. ƒ(x) = 2x3 + 5x2 º 6x º 1

17. ƒ(x) = 2x5 + x2 + 16 18. ƒ(x) = 2x3 + 9x2 º 53x º 60

19. ƒ(x) = 6x4 º 3x3 + x + 10 20. ƒ(x) = 4x3 + 5x2 º 3

21. ƒ(x) = 8x2 º 12x º 3 22. ƒ(x) = 3x4 + 2x3 º x + 15

USING SYNTHETIC DIVISION Use synthetic division to decide which of the

following are zeros of the function: 1, º1, 2, º2.

23. ƒ(x) = x3 + 7x2 º 4x º 28 24. ƒ(x) = x3 + 5x2 + 2x º 8

25. ƒ(x) = x4 + 3x3 º 7x2 º 27x º 18 26. ƒ(x) = 2x4 º 9x3 + 8x2 + 9x º 10

27. ƒ(x) = x4 + 3x3 + 3x2 º 3x º 4 28. ƒ(x) = 3x4 + 3x3 + 2x2 + 5x º 10

29. ƒ(x) = x3 º 3x2 + 4x º 12 30. ƒ(x) = x3 + x2 º 11x + 10

31. ƒ(x) = x6 º 2x4 º 11x2 + 12 32. ƒ(x) = x5 º x4 º 2x3 º x2 + x + 2


GUIDED PRACTICE


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Example 1: Exs. 15–32Example 2: Exs. 33–58Example 3: Exs. 59–64

¹ º » ¼ ½ ¾ ¿ ÀFINDING REAL ZEROS Find all the real zeros of the function.

33. ƒ(x) = x3 º 8x2 º 23x + 30 34. ƒ(x) = x3 + 2x2 º 11x º 12

35. ƒ(x) = x3 º 7x2 + 2x + 40 36. ƒ(x) = x3 + x2 º 2x º 2

37. ƒ(x) = x3 + 72 º 5x2 º 18x 38. ƒ(x) = x3 + 9x2 º 4x º 36

39. ƒ(x) = x4 º 5x3 + 7x2 + 3x º 10 40. ƒ(x) = x4 + x3 + x2 º 9x º 10

41. ƒ(x) = x4 + x3 º 11x2 º 9x + 18 42. ƒ(x) = x4 º 3x3 + 6x2 º 2x º 12

43. ƒ(x) = x5 + x4 º 9x3 º 5x2 º 36 44. ƒ(x) = x5 º x4 º 7x3 + 11x2 º 8x + 12

ELIMINATING POSSIBLE ZEROS Use the graph to shorten the list of possible

rational zeros. Then find all the real zeros of the function.

45. ƒ(x) = 4x3 º 12x2 º x + 15 46. ƒ(x) = º3x3 + 20x2 º 36x + 16

FINDING REAL ZEROS Find all the real zeros of the function.

47. ƒ(x) = 2x3 + 4x2 º 2x º 4 48. ƒ(x) = 2x3 º 5x2 º 14x + 8

49. ƒ(x) = 2x3 º 5x2 º x + 6 50. ƒ(x) = 2x3 + x2 º 50x º 25

51. ƒ(x) = 2x3 º x2 º 32x + 16 52. ƒ(x) = 3x3 + 12x2 + 3x º 18

53. ƒ(x) = 2x4 + 3x3 º 3x2 + 3x º 5 54. ƒ(x) = 3x4 º 8x3 º 5x2 + 16x º 5

55. ƒ(x) = 2x4 + x3 º x2 º x º 1 56. ƒ(x) = 3x4 + 11x3 + 11x2 + x º 2

57. ƒ(x) = 2x5 + x4 º 32x º 16 58. ƒ(x) = 3x5 + x4 º 243x º 81

59. HEALTH PRODUCT SALES From 1990 to 1994, the mail order sales ofhealth products in the United States can be modeled by

S = 10t3 + 115t2 + 25t + 2505

where S is the sales (in millions of dollars) and t is the number of years since 1990. In what year were about $3885 million of health products sold? (Hint: First substitute 3885 for S, then divide both sides by 5.)

60. MONUMENTS You are designing a monument and abase as shown at the right. You will use 90 cubic feet ofconcrete for both pieces. Find the value of x.

61. MOLTEN GLASS At a factory, molten glass is poured into molds to make paperweights. Each mold is a rectangular prism whose height is 3 inches greater than the length of each side of the square base. A machine pours 20 cubic inches of liquid glass into each mold. What are the dimensions of the mold?

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x ftx ft

0.5 ft

xÏ2 ft

xÏ2 ft

3x ft


Ex. 60

MOLTEN GLASS

In order for glass tomelt so that it can be pouredinto a mold, it must beheated to temperaturesbetween 1000°C and 2000°C.

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62. SAND CASTLES You are designing a kit formaking sand castles. You want one of the molds tobe a cone that will hold 48π cubic inches of sand.What should the dimensions of the cone be if youwant the height to be 5 inches more than the radiusof the base?

63. SWIMMING POOLS You are designing an in-ground lap swimming pool with a volume of2000 cubic feet. The width of the pool should be 5 feet more than the depth, and the length shouldbe 35 feet more than the depth. What should thedimensions of the pool be?

64. WHEELCHAIR RAMPS You are building a solid concrete wheelchair ramp.The width of the ramp is three times the height, and the length is 5 feet morethan 10 times the height. If 150 cubic feet of concrete is used, what are thedimensions of the ramp?

QUANTITATIVE COMPARISON In Exercises 65 and 66, choose the statement

that is true about the given quantities.

¡A The quantity in column A is greater.

¡B The quantity in column B is greater.

¡C The two quantities are equal.

¡D The relationship cannot be determined from the given information.

65.

66.

Find the real zeros of the function. Then match each function with its graph.

67. ƒ(x) = x3 + 2x2 º x º 2 68. g(x) = x3 º 3x + 2 69. h(x) = x3 º x2 + 2

A. B. C.

70. CRITICAL THINKING Is it possible for a cubic function to have more than threereal zeros? Is it possible for a cubic function to have no real zeros? Explain.

ÎÏ ÐÑÒ ÐÑÎÐÎÑÎ

x

3x

10x 1 5

x 1 5

x

x

x 1 35

x 1 5

TestPreparation

★★Challenge

Column A Column B

The number of possible rational zeros of The number of possible rational zeros of

ƒ(x) = x4 º 3x2 + 5x + 12 ƒ(x) = x2 º 13x + 20

The greatest real zero of The greatest real zero of

ƒ(x) = x3 + 2x2 º 5x º 6 ƒ(x) = x4 + 3x3 º 2x2 º 6x + 4

SAND SCULPTURE

The tallest sand

sculptures built were over

20 feet tall and each

consisted of hundreds

of tons of sand.

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EXTRA CHALLENGE




SOLVING QUADRATIC EQUATIONS Solve the equation. (Review 5.2 for 6.7)

71. x2 º 6x + 9 = 0 72. x2 + 12 = 10x º 13

73. x º 1 = x2 º x 74. x2 + 18 = 12x º x2

75. 2x2 º 20x = x2 º 100 76. x2 º 12x + 49 = 6x º 32

WRITING QUADRATIC FUNCTIONS Write a quadratic function in intercept

form whose graph has the given x-intercepts and passes through the given

point. (Review 5.8 for 6.7)

77. x-intercepts: º3, 3 78. x-intercepts: º5, 1 79. x-intercepts: º1, 5point: (0, 5) point: (º2, º6) point: (0, 10)

80. x-intercepts: 12, 7 81. x-intercepts: º12, º6 82. x-intercepts: 2, 8point: (º11, 7) point: (9, º5) point: (3, º4)

83. x-intercepts: 4, 10 84. x-intercepts: º6, 0 85. x-intercepts: º9, º1point: (7, 3) point: (2, 16) point: (1, 20)

86. PICTURE FRAMES You have a picture that you want to frame, but first youhave to put a mat around it. The picture is 12 inches by 16 inches. The area of themat is 204 square inches. If the mat extends beyond the picture the same amountin each direction, what will the final dimensions of the picture and mat be?(Review 5.2)

Factor the polynomial. (Lesson 6.4)

1. 5x3 + 135 2. 6x3 + 12x2 + 12x + 24

3. 4x5 º 16x 4. 3x3 º x2 º 15x + 5

Find the real-number solutions of the equation. (Lesson 6.4)

5. 7x4 = 252x2 6. 16x6 = 54x3

7. 6x5 º 18x4 + 12x3 = 36x2 8. 2x3 + 5x2 = 8x + 20

Divide. Use synthetic division when possible. (Lesson 6.5)

9. (x2 + 7x º 44) ÷ (x º 4) 10. (3x2 º 8x + 20) ÷ (3x + 2)

11. (4x3 º 7x2 º x + 10) ÷ (x2 º 3) 12. (12x4 + 5x3 + 3x2 º 5) ÷ (x + 1)

13. (x4 + 2x2 + 3x + 6) ÷ (x3 º 3) 14. (5x4 + 2x3 º x º 5) ÷ (x + 5)

Find all the real zeros of the function. (Lesson 6.6)

15. ƒ(x) = x3 º 4x2 º 7x + 28 16. ƒ(x) = x3 º 6x2 + 21x º 26

17. ƒ(x) = 2x3+ 15x2 + 22x º 15 18. ƒ(x) = 2x3 + 7x2 º 28x + 12

19. DESIGNING A PATIO You are a landscape artist designing a patio. Thesquare patio floor is to be made from 128 cubic feet of concrete. The thicknessof the floor is 15.5 feet less than each side length of the patio. What are thedimensions of the patio floor? (Lesson 6.6)


MIXED REVIEW


Use the

fundamental theorem of

algebra to determine the

number of zeros of a

polynomial function.

Use technology to

approximate the real zeros of

a polynomial function, as

applied in Example 5.

. To solve real-lifeproblems, such as finding

the American Indian, Aleut,

and Eskimo population in

Ex. 59.


GOAL 2

GOAL 1



Using the FundamentalTheorem of Algebra

THE FUNDAMENTAL THEOREM OF ALGEBRA

The following important theorem, called the fundamental theorem of algebra, was first proved by the famous German mathematician Carl Friedrich Gauss(1777–1855).

In the following activity you will investigate how the number of solutions of ƒ(x) = 0is related to the degree of the polynomial ƒ(x).

The equation x3 º 6x2 º 15x + 100 = 0, which can be written as (x + 4)(x º 5)2 = 0,has only two distinct solutions: º4 and 5. Because the factor x º 5 appears twice,however, you can count the solution 5 twice. So, with 5 counted as a

this third-degree equation can be said to have three solutions: º4, 5, and 5.

In general, when all real and imaginary solutions are counted (with all repeatedsolutions counted individually), an nth-degree polynomial equation has exactly n

solutions. Similarly, any nth-degree polynomial function has exactly n zeros.

Finding the Number of Solutions or Zeros

a. The equation x3 + 3x2 + 16x + 48 = 0 has three solutions: º3, 4i, and º4i.

b. The function ƒ(x) = x4 + 6x3 + 12x2 + 8x has four zeros: º2, º2, º2, and 0.

E X A M P L E 1

solution,

repeated

GOAL 1

6.7

Investigating the Number of Solutions

Solve each polynomial equation. State how many solutions the equationhas, and classify each as rational, irrational, or imaginary.

a. 2x º 1 = 0 b. x2 º 2 = 0 c. x3 º 1 = 0

Make a conjecture about the relationship between the degree of a polynomialƒ(x) and the number of solutions of ƒ(x) = 0.

Solve the equation x3 + x2 º x º 1 = 0. How many different solutions arethere? How can you reconcile this number with your conjecture?

2

1

DevelopingConcepts

ACTIVITY

If ƒ(x) is a polynomial of degree n where n > 0, then the equation ƒ(x) = 0 has

at least one root in the set of complex numbers.

THE FUNDAMENTAL THEOREM OF ALGEBRA

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Ó Ô Õ Ö × Ø Ù Ú

6.7 Using the Fundamental Theorem of Algebra 367

Finding the Zeros of a Polynomial Function

Find all the zeros of ƒ(x) = x5 º 2x4 + 8x2 º 13x + 6.

SOLUTION

The possible rational zeros are ±1, ±2, ±3, and ±6. Using synthetic division, youcan determine that 1 is a repeated zero and that º2 is also a zero. You can write thefunction in factored form as follows:

ƒ(x) = (x º 1)(x º 1)(x + 2)(x2 º 2x + 3)

Complete the factorization, using the quadratic formula to factor the trinomial.

ƒ(x) = (x º 1)(x º 1)(x + 2)[x º (1 + iÏ2w)][x º (1 º iÏ2w)]

c This factorization gives the following five zeros:

1, 1, º2, 1 + iÏ2w, and 1 º iÏ2w

The graph of ƒ is shown at the right. Note that onlythe real zeros appear as x-intercepts. Also note thatthe graph only touches the x-axis at the repeated zero x = 1, but crosses the x-axis at the zero x = º2.

. . . . . . . . . .

The graph in Example 2 illustrates the behavior of the graph of a polynomial functionnear its zeros. When a factor x º k is raised to an odd power, the graph crosses the x-axis at x = k. When a factor x º k is raised to an even power, the graph is tangentto the x-axis at x = k.

In Example 2 the zeros 1 + iÏ2w and 1 º iÏ2w are complex conjugates. The complexzeros of a polynomial function with real coefficients always occur in complexconjugate pairs. That is, if a + bi is a zero, then a º bi must also be a zero.

Using Zeros to Write Polynomial Functions

Write a polynomial function ƒ of least degree that has real coefficients, a leadingcoefficient of 1, and 2 and 1 + i as zeros.

SOLUTION

Because the coefficients are real and 1 + i is a zero, 1 º i must also be a zero. Usethe three zeros and the factor theorem to write ƒ(x) as a product of three factors.

ƒ(x) = (x º 2)[x º (1 + i)][x º (1 º i)] Write ƒ(x) in factored form.

= (x º 2)[(x º 1) º i][(x º 1) + i] Regroup terms.

= (x º 2)f(x º 1)2 º i2g Multiply.

= (x º 2)[x2 º 2x + 1 º (º1)] Expand power and use i 2 = º1.

= (x º 2)(x2 º 2x + 2) Simplify.

= x3 º 2x2 + 2x º 2x2 + 4x º 4 Multiply.

= x3 º 4x2 + 6x º 4 Combine like terms.

✓CHECK You can check this result by evaluating ƒ(x) at each of its three zeros.

E X A M P L E 3

E X A M P L E 2

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ß à á â ã ä å æ


USING TECHNOLOGY TO APPROXIMATE ZEROS

The rational zero theorem gives you a way to find the rational zeros of a polynomialfunction with integer coefficients. To find the real zeros of any polynomial function,you may need to use technology.

Approximating Real Zeros

Approximate the real zeros of ƒ(x) = x4 º 2x3 º x2 º 2x º 2.

SOLUTION

There are several ways to use a graphing calculator to approximate the real zeros of afunction. One way is to use the Zero (or Root) feature as shown below.

c From these screens, you can see that the real zeros are about º0.73 and 2.73.

Because the polynomial function has degree 4, you know that there must be twoother zeros. These may be repeats of the real zeros, or they may be imaginary zeros.In this particular case, the two other zeros are imaginary: x = ±i.

Approximating Real Zeros of a Real-Life Function

PHYSIOLOGY For one group of people it was found that a person’s score S on theHarvard Step Test was related to his or her amount of hemoglobin x (in grams per100 milliliters of blood) by the following model:

S = º0.015x3 + 0.6x2 º 2.4x + 19

The normal range of hemoglobin is 12–18 grams per 100 milliliters of blood.Approximate the amount of hemoglobin for a person who scored 75.

SOLUTION

You can solve the equation

75 = º0.015x3 + 0.6x2 º 2.4x + 19

by rewriting it as 0 = º0.015x3 + 0.6x2 º 2.4x º 56and then using a graphing calculator to approximate thereal zeros of ƒ(x) = º0.015x3 + 0.6x2 º 2.4x º 56.From the graph you can see that there are three realzeros: x ≈ º7.3, x ≈ 16.4, and x ≈ 30.9.

c The person’s hemoglobin is probably about 16.4 grams per 100 milliliters ofblood, since this is the only zero within the normal range.

E X A M P L E 5

E X A M P L E 4

GOAL 2

Zero

X=16.428642 Y=0

Zero

X=-.7320508 Y=0

Zero

X=2.7320508 Y=0

HARVARD STEP

TEST When taking the Harvard Step Test,a person steps up and downa 20 inch platform for5 minutes. The person’sscore is determined by hisor her heart rate in the firstfew minutes after stopping.

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1. State the fundamental theorem of algebra.

2. Two zeros of ƒ(x) = x3 º 6x2 º 16x + 96 are 4 andº4. Explain why the third zero must also be a realnumber.

3. The graph of ƒ(x) = x3 º x2 º 8x + 12 is shownat the right. How many real zeros does the functionhave? How many imaginary zeros does the functionhave? Explain your reasoning.

Find all the zeros of the polynomial function.

4. ƒ(x) = x3 º x2 º 2x 5. ƒ(x) = x4 + x2 º 12

6. ƒ(x) = x3 + 5x2 º 9x º 45 7. ƒ(x) = x4 º x3 + 2x2 º 4x º 8

Write a polynomial function of least degree that has real coefficients, the given

zeros, and a leading coefficient of 1.

8. 3, 0, º2 9. 1, 1, i, ºi 10. 5, 2 + 3i

11. 1, º1, 2, º2, 3 12. 3, º2, º1 + i 13. 4i, 4i

14. GROCERY STORE REVENUE For the 25 years that a grocery store hasbeen open, its annual revenue R (in millions of dollars) can be modeled by

R = }

10,1000}(ºt4 + 12t3 º 77t2 + 600t + 13,650)

where t is the number of years the store has been open. In what year(s) was therevenue $1.5 million?

CHECKING ZEROS Decide whether the given x-value is a zero of the function.

15. ƒ(x) = x3 º x2 + 4x º 4, x = 1 16. ƒ(x) = x3 + 3x2 º 5x + 8, x = 4

17. ƒ(x) = x4 º x2 º 3x + 3, x = 0 18. ƒ(x) = x3 + 5x2 + x + 5, x = º5

19. ƒ(x) = x3 º 4x2 + 16x º 64, x = 4i 20. ƒ(x) = x3 º 3x2 + x º 3, x = ºi

FINDING ZEROS Find all the zeros of the polynomial function.

21. ƒ(x) = x4 + 5x3 + 5x2 º 5x º 6 22. ƒ(x) = x4 + 4x3 º 6x2 º 36x º 27

23. ƒ(x) = x3 º 4x2 + 3x 24. ƒ(x) = x3 + 5x2 º 4x º 20

25. ƒ(x) = x4 + 7x3 º x2 º 67x º 60 26. ƒ(x) = x4 º 5x2 º 36

27. ƒ(x) = x3 º x2 + 49x º 49 28. ƒ(x) = x3 º x2 + 25x º 25

29. ƒ(x) = x4 + 6x3 + 14x2 + 54x + 45 30. ƒ(x) = x3 + 3x2 + 25x + 75

31. ƒ(x) = x4 º x3 º 5x2 º x º 6 32. ƒ(x) = x4 + x3 + 2x2 + 4x º 8

33. ƒ(x) = 2x4 º 7x3 º 27x2 + 63x + 81 34. ƒ(x) = 2x4 º x3 º 42x2 + 16x + 160



Skill Check ✓

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Example 1: Exs. 21–54Example 2: Exs. 21–34Example 3: Exs. 35–46Example 4: Exs. 47–54Example 5: Exs. 55–59

GUIDED PRACTICE

Ex. 3



WRITING POLYNOMIAL FUNCTIONS Write a polynomial function of least

degree that has real coefficients, the given zeros, and a leading coefficient of 1.

35. 2, 1, 4 36. 1, º4, 5 37. º6, 3, 5

38. º5, 2, º2 39. º2, º4, º7 40. 8, ºi, i

41. 3i, º3i, 5 42. 2, º2, º6i 43. i, º3i, 3i

44. 3 º i, 5i 45. 4, 4, 2 + i 46. º2, º2, 3, º4i

FINDING ZEROS Use a graphing calculator to graph the polynomial

function. Then use the Zero (or Root) feature of the calculator to find

the real zeros of the function.

47. ƒ(x) = x3 º x2 º 5x + 3 48. ƒ(x) = 2x3 º x2 º 3x º 1

49. ƒ(x) = x3 º 2x2 + x + 1 50. ƒ(x) = x4 º 2x º 1

51. ƒ(x) = x4 º x3 º 4x2 º 3x º 2 52. ƒ(x) = x4 º x3 º 3x2 º x + 1

53. ƒ(x) = x4 + 3x2 º 2 54. ƒ(x) = x4 º x3 º 20x2 + 10x + 27

GRAPHING MODELS In Exercises 55–59, you may find it helpful to graph

the model on a graphing calculator.

55. UNITED STATES EXPORTS For 1980 through 1996, the total exports E (in billions of dollars) of the United States can be modeled by

E = º0.131t3 + 5.033t2 º 23.2t + 233

where t is the number of years since 1980. In what year were the total exports about$312.76 billion? c Source: U.S. Bureau of the Census

56. EDUCATION DONATIONS For 1983 through 1995, the amount of privatedonations D (in millions of dollars) allocated to education can be modeled by

D = 1.78t3 º 6.02t2 + 752t + 6701

where t is the number of years since 1983. In what year was $14.3 billion of privatedonations allocated to education? c Source: AAFRC Trust for Philanthropy

57. SPORTS EQUIPMENT For 1987 through 1996, the sales S (in millions ofdollars) of gym shoes and sneakers can be modeled by

S = º0.982t5 + 24.6t4 º 211t3 + 661t2 º 318t + 1520

where t is the number of years since 1987. Were there any years in which saleswere about $2 billion? Explain. c Source: National Sporting Goods Association

58. TELEVISION For 1990 through 2000, the actual and projected amount spenton television per person per year in the United States can be modeled by

S = º0.213t3 + 3.96t2 + 10.2t + 366

where S is the amount spent (in dollars) and t is the number of years since 1990.During which year was $455 spent per person on television?

c Source: Veronis, Suhler & Associates, Inc.

59. POPULATION For 1890 through 1990, the American Indian, Eskimo, andAleut population P (in thousands) can be modeled by the function

P = 0.00496t3 º 0.432t2 + 11.3t + 212

where t is the number of years since 1890. In what year did the population reach722,000?

FOCUS ON

APPLICATIONS

UNITED STATES

EXPORTS TheUnited States exports morethan any other country inthe world. It also importsmore than any othercountry.

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DATA UPDATE of Statistical Abstract of the United States data at www.mcdougallittell.com INT

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ß à á â ã ä å æ60. MULTI-STEP PROBLEM Mary plans to save $1000 each summer to buy a used

car at the end of the fourth summer. At the end of each summer, she will depositthe $1000 she earned from her summer job into her bank account. The tableshows the value of her deposits over the four year period. In the table, g is thegrowth factor 1 + r where r is the annual interest rate expressed as a decimal.

a. Copy and complete the table.

b. Write a polynomial function of g that represents the value of Mary’s accountat the end of the fourth summer.

c. Writing Suppose Mary wants to buy a car that costs about $4300. Whatgrowth factor does she need to obtain this amount? What annual interest ratedoes she need? Explain how you found your answers.

61. a. Copy and complete the table.

b. Use your completed table to make a conjecture relating the sum of the zerosof a polynomial function with the coefficients of the polynomial function.

c. Use your completed table to make a conjecture relating the product of the zerosof a polynomial function with the coefficients of the polynomial function.

62. Show that the sum of a pair of complex conjugates is a real number.

63. Show that the product of a pair of complex conjugates is a real number.

GRAPHING WITH INTERCEPT FORM Graph the quadratic function. Label the

vertex, axis of symmetry, and x-intercepts. (Review 5.1 for 6.8)

64. y = º3(x º 2)(x + 2) 65. y = 2(x º 1)(x º 5)

66. y = 2(x + 4)(x º 3) 67. y = º(x + 1)(x º 5)

GRAPHING POLYNOMIALS Graph the polynomial function. (Review 6.2 for 6.8)

68. ƒ(x) = º2x4 69. ƒ(x) = ºx3 º 4

70. ƒ(x) = x3 + 4x º 3 71. ƒ(x) = x4 º 3x3 + x + 2

MIXED REVIEW

End of End of End of End of1st summer 2nd summer 3rd summer 4th summer

Value of 1st deposit 1000 1000g 1000g2 1000g3

Value of 2nd deposit___

1000 ? ?

Value of 3rd deposit___ ___

1000 ?

Value of 4th deposit___ ___ ___

1000


TestPreparation

★★Challenge

Function Zeros Sum of zeros Product of zeros

ƒ(x) = x2 º 5x + 6 ? ? ?

ƒ(x) = x3 º 7x + 6 ? ? ?

ƒ(x) = x4 + 2x3 + x2 + 8x º 12 ? ? ?

ƒ(x) = x5 º 3x4 º 9x3 + 25x2 º 6x ? ? ?

EXTRA CHALLENGE


ç è é ê ë ì í îSolving Polynomial Equations

In Lesson 6.4 you learned to solve polynomial equations by factoring. When

factoring is not possible, you can use a graphing calculator instead.

c EXAMPLE

Use a graphing calculator to find the real solutions of x3 + 4x2 º 2x + 5 = 19.

c SOLUTION

c The solutions are x ≈ º3.35, x ≈ º2.40, and x ≈ 1.74.

c EXERCISES

Use a graphing calculator to find the real solutions of the equation.

1. }

12

}x3 º 3x2 + x + 6 = 4 2. 2x3 º 8x2 + 5x + 14 = 7

3. x3 º 5x2 + x + 3 = 8 4. 0.3x3 º 5x2 + 8x + 15 = 13

5. x4 º 6x2 + 5 = 2 6. 0.2x4 º 3x3 º 12x2 + 8x + 22 = 13

7. 17x5 º 24x3 + x2 + 2x = 4 8. º1.25x5 + 3.75x2 + 0.4x º 6 = º4

9. Look back at Example 5 on page 368. Use the method described above to solve

the problem. Does your answer agree with the answer given in Example 5?


Using Technology

Graphing Calculator Activity for use with Lesson 6.7 ACTIVITY 6.7

Intersection

X=-3.348894 Y=19

STUDENT HELP

KEYSTROKE

HELP

See keystrokes for several models ofcalculators atwww.mcdougallittell.com

INT

ERNET

To solve the equation

graphically, graph each side

of the equation as follows.

1 When the equations are graphed

in the standard viewing window,

you see most of the graph of y1,

but none of the graph of y2.

2

The graphs of y1 and y2

intersect at three points. Use

the Intersect feature to find the

x-coordinates of these points.

4

Y1=X^3+4X2-2X+5

Y2=19

Y3=

Y4=

Y5=

Y6=

Y7=

You know that y2 = 19 is a

horizontal line. Change the

viewing window so that

º10 ≤ x ≤ 10 and º2 ≤ y ≤ 22.

3

ç è é ê ë ì í î

6.8 Analyzing Graphs of Polynomial Functions 373

Analyzing Graphs ofPolynomial Functions

ANALYZING POLYNOMIAL GRAPHS

In this chapter you have learned that zeros, factors, solutions, and x-intercepts areclosely related concepts. The relationships are summarized below.

Using x-Intercepts to Graph a Polynomial Function

Graph the function ƒ(x) = }

14

}(x + 2)(x º 1)2.

SOLUTION

Plot x-intercepts. Since x + 2 and x º 1 are factors of ƒ(x), º2 and 1 are thex-intercepts of the graph of ƒ. Plot the points (º2, 0) and (1, 0).

Plot points between and beyond the x-intercepts.

Determine the end behavior of the graph. Because ƒ(x) has three linear factors of the form x º k and a constant factor of

}

14

}, it is a cubic function with a positive

leading coefficient. Therefore, ƒ(x) ˘ º‡ as x ˘ º‡ and ƒ(x) ˘ +‡ as x ˘ +‡.

Draw the graph so that it passes through the points you plotted and has the appropriate endbehavior.

E X A M P L E 1

GOAL 1

Analyze the graph

of a polynomial function.

Use the graph of

a polynomial function to

answer questions about

real-life situations, such as

maximizing the volume of a

box in Example 3.

. To find the maximum and

minimum values of real-lifefunctions, such as the

function modeling orange

consumption in the United

States in Ex. 36.


GOAL 2

GOAL 1


6.8

Let ƒ(x) = anxn + an º 1xn º 1 + . . . + a1x + a0 be a polynomial function.

The following statements are equivalent.

ZERO: k is a zero of the polynomial function ƒ.

FACTOR: x º k is a factor of the polynomial ƒ(x).

SOLUTION: k is a solution of the polynomial equation ƒ(x) = 0.

If k is a real number, then the following is also equivalent.

X-INTERCEPT: k is an x-intercept of the graph of the polynomial function ƒ.

ZEROS, FACTORS, SOLUTIONS, AND INTERCEPTSCONCEPT

SUMMARY

x º4 º3 º1 0 2 3

y º12}

12

} º4 1 }

12

} 1 5 ïðñ(22, 0)

ò(1, 0)

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TURNING POINTS Another important characteristic of graphs of polynomial functions is that they have turning points corresponding to local maximum andminimum values. The y-coordinate of a turning point is a

of the function if the point is higher thanall nearby points. The y-coordinate of a turning point is a

if the point is lower than all nearby points.

Recall that in Chapter 5 you used technology to find the maximums and minimumsof quadratic functions. In Example 2 you will use technology to find turning points of higher-degree polynomial functions. If you take calculus, you will learn symbolictechniques for finding maximums and minimums.

Finding Turning Points

Graph each function. Identify the x-intercepts and the points where the local

maximums and local minimums occur.

a. ƒ(x) = x3 º 3x2 + 2 b. ƒ(x) = x4 º 4x3 º x2 + 12x º 2

SOLUTION

a. Use a graphing calculator to graph the function.

Notice that the graph has three x-intercepts and twoturning points. You can use the graphing calculator’sZero, Maximum, and Minimum features to approximatethe coordinates of the points.

c The x-intercepts of the graph are x ≈ º0.73, x = 1,and x ≈ 2.73. The function has a local maximum at(0, 2) and a local minimum at (2, º2).

b. Use a graphing calculator to graph the function.

Notice that the graph has four x-intercepts and threeturning points. You can use the graphing calculator’sZero, Maximum, and Minimum features to approximatethe coordinates of the points.

c The x-intercepts of the graph are x ≈ º1.63, x ≈ 0.17, x ≈ 2.25, and x ≈ 3.20. The function has local minimums at (º0.94, º10.06) and (2.79, º2.58), and it has a local maximum at (1.14, 6.14).

E X A M P L E 2

local minimum

local maximum

ï ðlocalmaximum

localminimum

Maximum

X=0 Y=2

Minimum

X=-.9385361 Y=-10.06055

The graph of every polynomial function of degree n has at most n º 1

turning points. Moreover, if a polynomial function has n distinct real zeros,

then its graph has exactly n º 1 turning points.

TURNING POINTS OF POLYNOMIAL FUNCTIONS

HOMEWORK HELP


INT

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STUDENT HELP

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USING POLYNOMIAL FUNCTIONS IN REAL LIFE

In the following example, technology is used to maximize a polynomial function thatmodels a real-life situation.

Maximizing a Polynomial Model

You are designing an open box to be made of a piece of cardboard that is 10 inchesby 15 inches. The box will be formed by making the cuts shown in the diagram andfolding up the sides so that the flaps are square. You want the box to have the greatestvolume possible. How long should you make the cuts? What is the maximumvolume? What will the dimensions of the finished box be?

SOLUTION

To find the maximum volume, graph the volumefunction on a graphing calculator as shown at theright. When you use the Maximum feature, youconsider only the interval 0 < x < 5 because thisdescribes the physical restrictions on the size of theflaps. From the graph, you can see that the maximumvolume is about 132 and occurs when x ≈ 1.96.

c You should make the cuts approximately 2 inches long. The maximum volume is about 132 cubic inches. The dimensions of the box with this volume will be x = 2 inches by 10 º 2x = 6 inches by 15 º 2x = 11 inches.

E X A M P L E 3

GOAL 2

RE

AL LIFE

RE

AL LIFE

Manufacturing

VERBAL

MODEL

ALGEBRAIC

MODEL

PROBLEMSOLVING

STRATEGY

LABELS

= • •

Volume = (cubic inches)

Width = (inches)

Length = (inches)

Height = (inches)

=

= (4x2 º 50x + 150)x

= 4x3 º 50x2 + 150x

x(15 º 2x)(10 º 2x)V

x

15 º 2x

10 º 2x

V

HeightLengthWidthVolume

10 in.

15 in.

x

x

x

x

x

x

x

x

Maximum

X=1.9618749 Y=132.03824

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1. Explain what a local maximum of a function is.

2. Let ƒ be a fourth-degree polynomial function with these zeros: 6, º2, 2i, and º2i.

a. How many distinct linear factors does ƒ(x) have?

b. How many distinct solutions does ƒ(x) = 0 have?

c. What are the x-intercepts of the graph of ƒ?

3. Let ƒ be a fifth-degree polynomial function with five distinct real zeros. Howmany turning points does the graph of ƒ have?

Graph the function.

4. ƒ(x) = (x º 1)(x + 3)2 5. ƒ(x) = (x º 1)(x + 1)(x º 3)

6. ƒ(x) = }

18

}(x + 1)(x º 1)(x º 3) 7. ƒ(x) = }

15

}(x º 3)2(x + 1)2

Use a graphing calculator to graph the function. Identify the x-intercepts

and the points where the local maximums and local minimums occur.

8. ƒ(x) = 3x4 º 5x2 + 2x + 1 9. ƒ(x) = x3 º 3x2 + x + 1

10. ƒ(x) = º2x3 + x2 + 4x 11. ƒ(x) = x5 + x4 º 4x3 º 3x2 + 5x

12. MANUFACTURING In Example 3, supposeyou used a piece of cardboard that is 18 inches by 18 inches. Then the volume of the box wouldbe given by this function:

V = 4x3 º 72x2 + 324x

Using a graphing calculator, you would obtainthe graph shown at the right.

a. What is the domain of the volume function? Explain.

b. Use the graph to estimate the length of the cut that will maximize the volumeof the box.

c. Estimate the maximum volume the box can have.

GRAPHING POLYNOMIAL FUNCTIONS Graph the function.

13. ƒ(x) = (x º 1)3(x + 1) 14. ƒ(x) = }

110}(x + 3)(x º 1)(x º 4)

15. ƒ(x) = }

18

}(x + 4)(x + 2)(x º 3) 16. ƒ(x) = 2(x + 2)2(x + 4)2

17. ƒ(x) = 5(x º 1)(x º 2)(x º 3) 18. ƒ(x) = }

112}(x + 4)(x º 3)(x + 1)2

19. ƒ(x) = (x + 1)(x2 º 3x + 3) 20. ƒ(x) = (x + 2)(2x2 º 2x + 1)

21. ƒ(x) = (x º 2)(x2 + x + 1) 22. ƒ(x) = (x º 3)(x2 º x + 1)


GUIDED PRACTICE


Skill Check ✓

0

22 0 8 10642

500

400

300

200

100

STUDENT HELP


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ANALYZING GRAPHS Estimate the coordinates of each turning point and state

whether each corresponds to a local maximum or a local minimum. Then list all

the real zeros and determine the least degree that the function can have.

23. 24. 25.

26. 27. 28.

USING GRAPHS Use a graphing calculator to graph the polynomial

function. Identify the x-intercepts and the points where the local

maximums and local minimums occur.

29. ƒ(x) = 3x3 º 9x + 1 30. ƒ(x) = º}

13

}x3 + x º }

32

}

31. ƒ(x) = º}

14

}x4 + 2x2 32. ƒ(x) = x5 º 6x3 + 9x

33. ƒ(x) = x5 º 5x3 + 4x 34. ƒ(x) = x4 º 2x3 º 3x2 + 5x + 2

35. SWIMMING The polynomial function

S = º241t7 + 1062t6 º 1871t5 + 1647t4 º 737t3 + 144t2 º 2.432t

models the speed S (in meters per second) of a swimmer doing the breast strokeduring one complete stroke, where t is the number of seconds since the start ofthe stroke. Graph the function. At what time is the swimmer going the fastest?

36. FOOD The average amount of oranges (in pounds) eaten per person eachyear in the United States from 1991 to 1996 can be modeled by

ƒ(x) = 0.298x3 º 2.73x2 + 7.05x + 8.45

where x is the number of years since 1991. Graph the function and identify any turning points on the interval 0 ≤ x ≤ 5. What real-life meaning do thesepoints have?

QUONSET HUTS In Exercises 37º39, use the following information.

A quonset hut is a dwelling shaped like half a cylinder. Suppose you have 600 squarefeet of material with which to build a quonset hut.

37. The formula for surface area is S = πr2 + πrl where r is the radius of thesemicircle and l is the length of the hut. Substitute 600 for S and solve for l.

38. The formula for the volume of the hut is V = }

12

}πr2l. Write an equation for the

volume V of the quonset hut as a polynomial function of r by substituting the expression for l from Exercise 37 into the volume formula.

39. Use the function you wrote in Exercise 38 to find the maximum volume of a quonset hut with a surface area of 600 square feet. What are the hut’sdimensions?

� ��2

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�� STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 13–22Example 2: Exs. 23–34Example 3: Exs. 35–40

QUONSET HUTS

were invented duringWorld War II. They weretemporary structures thatcould be assembled quicklyand easily. After the warthey were sold as homes forabout $1000 each.

APPLICATION LINK


INT

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RE

AL LIFE

RE

AL LIFE

FOCUS ON

APPLICATIONS

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40. CONSUMER ECONOMICS The producer price index of butter from 1991 to 1997 can be modeled by P = º0.233x4 + 2.64x3 º 6.59x2 º 3.93x + 69.1where x is the number of years since 1991. Graph the function and identify any turning points on the interval 0 ≤ x ≤ 6. What real-life meaning do thesepoints have?

41. CRITICAL THINKING Sketch the graph of a polynomial function that has threeturning points. Label each turning point as a local maximum or local minimum.What must be true about the degree of the polynomial function that has such agraph? Explain your reasoning.

In Exercises 42 and 43, use the graph of the polynomial

function ƒ shown at the right.

42. MULTIPLE CHOICE What is the local maximum of ƒ on the interval º2 ≤ x ≤ º1?

¡A ƒ(x) ≈ 3.7 ¡B ƒ(x) ≈ 1.4

¡C ƒ(x) ≈ º1.4 ¡D ƒ(x) ≈ º3.7

43. MULTIPLE CHOICE What is the local maximum of ƒ on the interval º1 ≤ x ≤ 1?

¡A ƒ(x) ≈ 3.7 ¡B ƒ(x) ≈ 1.4 ¡C ƒ(x) ≈ º1.4 ¡D ƒ(x) ≈ º3.7

44. GRAPHING OPPOSITES Sketch the graph of y = ƒ(x) for this function:

ƒ(x) = x3 + 4x2

Then sketch the graph of y = ºƒ(x). Explain how the graphs, the x-intercepts, thelocal maximums, and the local minimums are related. Finally, sketch the graph of y = ƒ(ºx). Compare it with the others.

RELATING VARIABLES The variables x and y vary directly. Write an equation

that relates the variables. (Review 2.4)

45. x = 1, y = 7 46. x = º4, y = 6 47. x = 12, y = 3

48. x = 2, y = º5 49. x = º5, y = 3 50. x = º6, y = º15

MATRIX PRODUCTS Let A and B be matrices with the given dimensions. State

whether the product AB is defined. If so, give the dimensions of AB. (Review 4.2)

51. A: 4 ª 3, B: 3 ª 1 52. A: 2 ª 4, B: 4 ª 5

53. A: 4 ª 3, B: 2 ª 4 54. A: 6 ª 6, B: 6 ª 5

WRITING QUADRATIC FUNCTIONS Write a quadratic function whose graph

passes through the given points. (Review 5.8 for 6.9)

55. vertex: (1, 4); point: (4, º5) 56. vertex: (º2, 6); point: (0, 2)

57. points: (º5, 0), (5, 0), (7, 5) 58. points: (º2, 0), (4, 0), (1, º4)

59. PLANT GROWTH You have a kudzu vine in your back yard. On Monday,the vine is 30 inches long. The following Thursday, the vine is 60 inches long. What is the average rate of change in the length of the vine? (Lesson 2.2)

MIXED REVIEW

TestPreparation

★★Challenge

� ��

EXTRA CHALLENGE


� � � � �Developing Concepts

ACTIVITY 6.9Group Activity for use with Lesson 6.9

GROUP ACTIVITY

Work with a partner.

MATERIALS

• Paper

• Pencil

Exploring Finite Differences

c QUESTION How are the finite differences for a polynomial function related

to the function’s degree?

c EXPLORING THE CONCEPT

The number of paths that lead, through a sequence of upward and rightwardmovements only, from the bottom left corner of a grid to the top right corner depends on the grid’s dimensions.

For an n ª 1 grid, the number of paths is given by ƒ(n) = n + 1.

For an n ª 2 grid, the number of paths is given by g(n) = }

12

}(n + 1)(n + 2).

For an n ª 3 grid, the number of paths is given by h(n) = }

16

}(n + 1)(n + 2)(n + 3).

For each function given above, follow Steps 1–4. The steps for the first function ƒhave been done for you.

Evaluate the function when n = 1, 2, 3, 4, 5,and 6.

For each pair of consecutive function values, find the difference of the values. These numbers are called first-order differences.

Are the first-order differences constant? If so, stop here. Otherwise, find thedifferences in each pair of consecutive first-order differences. These numbers are called second-order differences.

Are the second-order differences constant? If so, stop here. Otherwise, find thedifferences in each pair of consecutive second-order differences. These numbersare called third-order differences.

c DRAWING CONCLUSIONS

1. Repeat Steps 1–4 for each function.

a. ƒ(n) = 3n + 1 b. ƒ(n) = n2 + 2n º 3 c. ƒ(n) = 2n3 º n + 4

2. For each function giving the number of paths through a grid and for eachfunction in Exercise 1, state the degree of the function and the number of timesdifferences were calculated before a row of constant, nonzero differences wasobtained. What do you notice?

3. Which order differences do you think will be constant for the functionƒ(n) = n4 + n? Explain. Then find the differences to see if you are correct.

4

3

2

1

n 1 2 3 4 n 1 2 3 4 n 1 2 3 4

ƒ(n) 5 n 1 1 g (n) 5 (n 1 1)(n 1 2)12

h (n) 5 (n 1 1)(n 1 2)(n 1 3)16

6.9 Concept Activity 379

n 1 2 3 4 5 6

f (n) 2 3 4 5 6 7

1 1 1 1 1

First-order differences

are constant.

� � � � � � � �


Modeling with Polynomial Functions

USING FINITE DIFFERENCES

You know that two points determine a line and that three points determine a parabola.In Example 1 you will see that four points determine the graph of a cubic function.

Writing a Cubic Function

Write the cubic function whose graph is shown at the right.

SOLUTION

Use the three given x-intercepts to write the following:

ƒ(x) = a(x + 3)(x º 2)(x º 5)

To find a, substitute the coordinates of the fourth point.

º15 = a(0 + 3)(0 º 2)(0 º 5), so a = º}

21

}

c ƒ(x) = º}

12

}(x + 3)(x º 2)(x º 5)

✓CHECK Check the graph’s end behavior. The degree of ƒ

is odd and a < 0, so ƒ(x) ˘ +‡ as x ˘ º‡ and ƒ(x) ˘ º‡ as x ˘ +‡.

. . . . . . . . . .

To decide whether y-values for equally-spaced x-values can be modeled by apolynomial function, you can use

Finding Finite Differences

The first three triangular numbers are shown at the right.

A formula for the nth triangular number is ƒ(n) = }

12

}(n2 + n).

Show that this function has constant second-order differences.

SOLUTION

Write the first several triangular numbers. Find the first-orderdifferences by subtracting consecutive triangular numbers. Then find the second-order differences by subtracting consecutive first-order differences.

E X A M P L E 2

finite differences.

E X A M P L E 1

GOAL 1

Use finite

differences to determine the

degree of a polynomial

function that will fit a set

of data.

Use technology to

find polynomial models for

real-life data, as applied in

Example 4.

. To model real-lifequantities, such as the speed

of the space shuttle

in Ex. 49.


GOAL 2

GOAL 1


6.9

E X P L O R I N G D ATA

A N D S TAT I S T I C S

� ��(–3, 0)

(2, 0)

(5, 0)�(0, 215)

REA

L LIFE

REA

L LIFE

ƒ(1) ƒ(2) ƒ(3) ƒ(4) ƒ(5) ƒ(6) ƒ(7)

1 3 6 10 15 21 28

2 3 4 5 6 7

1 1 1 1 1

Function values for

equally-spaced n-values


Second-order differences

ƒ(1) = 1

ƒ(2) = 3

ƒ(3) = 6

Triangular Numbers

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6.9 Modeling with Polynomial Functions 381

In Example 2 notice that the function has degree two and that the second-orderdifferences are constant. This illustrates the first property of finite differences.

The following example illustrates the second property of finite differences.

Modeling with Finite Differences

The first six triangular pyramidal numbers are shown below. Find a polynomialfunction that gives the nth triangular pyramidal number.

SOLUTION

Begin by finding the finite differences.

Because the third-order differences are constant, you know that the numbers can berepresented by a cubic function which has the form ƒ(n) = an3 + bn2 + cn + d.

By substituting the first four triangular pyramidal numbers into the function, you canobtain a system of four linear equations in four variables.

a(1)3 + b(1)2 + c(1) + d = 1 a + b + c + d = 1

a(2)3 + b(2)2 + c(2) + d = 4 8a + 4b + 2c + d = 4

a(3)3 + b(3)2 + c(3) + d = 10 27a + 9b + 3c + d = 10

a(4)3 + b(4)2 + c(4) + d = 20 64a + 16b + 4c + d = 20

Using a calculator to solve the system gives a = }

16

}, b = }

12

}, c = }

13

}, and d = 0.

c The nth triangular pyramidal number is given by ƒ(n) = }

16

}n3 + }

12

}n2 + }

13

}n.

E X A M P L E 3

1. If a polynomial function ƒ(x) has degree n, then the nth-order differences of

function values for equally spaced x-values are nonzero and constant.

2. Conversely, if the nth-order differences of equally-spaced data are nonzero

and constant, then the data can be represented by a polynomial function of

degree n.

PROPERTIES OF FINITE DIFFERENCES

f(1) f(2) f(3) f(4) f(5) f(6) f(7)

1 4 10 20 35 56 84

3 6 10 15 21 28

3 4 5 6 7

1 1 1 1

Function values for

equally-spaced n-values


Second-order differences

Third-order differences

ƒ(1) = 1 ƒ(2) = 4 ƒ(3) = 10 ƒ(4) = 20 ƒ(5) = 35 ƒ(6) = 56 ƒ(7) = 84

HOMEWORK HELP


INT

ERNET

STUDENT HELP

� � � � ! " #


POLYNOMIAL MODELING WITH TECHNOLOGY

In Examples 1 and 3 you found a cubic model that exactly fits a set of data points. In many real-life situations, you cannot find a simple model to fit data points exactly.Instead you can use the regression feature on a graphing calculator to find an nth-degree polynomial model that best fits the data.

Modeling with Cubic Regression

BOATING The data in the table give the average speed y (in knots) of the Trident

motor yacht for several different engine speeds x (in hundreds of revolutions perminute, or RPMs).

a. Find a polynomial model for the data.

b. Estimate the average speed of the Trident for an engine speed of 2400 RPMs.

c. What engine speed produces a boat speed of 14 knots?

SOLUTION

a. Enter the data in a graphing calculator andmake a scatter plot. From the scatter plot, itappears that a cubic function will fit the databetter than a linear or quadratic function.

Use cubic regression to obtain a model.

c y = 0.00475x3 º 0.194x2 + 3.13x º 9.53

✓CHECK By graphing the model in the sameviewing window as the scatter plot, you cansee that it is a good fit.

b. Substitute x = 24 into the model from part (a).

y = 0.00475(24)3 º 0.194(24)2 + 3.13(24) º 9.53

= 19.51

c The Trident’s speed for an engine speed of 2400 RPMs is about 19.5 knots.

c. Graph the model and the equation y = 14 onthe same screen. Use the Intersect feature tofind the point where the graphs intersect.

c An engine speed of about 2050 RPMsproduces a boat speed of 14 knots.

E X A M P L E 4

GOAL 2

Engine speed, x 9 11 13 15 17 19 21.5

Boat speed, y 6.43 7.61 8.82 9.86 10.88 12.36 15.24

Intersection

X=20.475611 Y=14

MOTORBOATS

often havetachometers instead ofspeedometers. Thetachometer measures theengine speed in revolutionsper minute, which can thenbe used to determine thespeed of the boat.

RE

AL LIFE

RE

AL LIFE

FOCUS ON

APPLICATIONS

� � � � ! " #1. Describe what first-order differences and second-order differences are.

2. How many points do you need to determine a quartic function?

3. Why can’t you use finite differences to find a model for the data in Example 4?

4. Write the cubic function whose graph passes through (3, 0), (º1, 0), (º2, 0),and (1, 2).

Show that the n th-order finite differences for the given function of degree n

are nonzero and constant.

5. ƒ(x) = 5x2 º 2x + 1 6. ƒ(x) = x3 + x2 º 1

7. ƒ(x) = x4 + 2x 8. ƒ(x) = 2x3 º 12x2 º 5x + 3

Use finite differences to determine the degree of the polynomial function that

will fit the data.

9. 10.

Find a polynomial function that fits the data.

11. 12.

13. Find a polynomial function that gives the number ofdiagonals of a polygon with n sides.

WRITING CUBIC FUNCTIONS Write the cubic function whose graph is shown.

14. 15. 16.

FINDING A CUBIC MODEL Write a cubic function whose graph passes

through the given points.

17. (º1, 0), (º2, 0), (0, 0), (1, º3) 18. (3, 0), (2, 0), (º3, 0), (1, º1)

19. (1, 0), (3, 0), (º2, 0), (2, 1) 20. (º1, 0), (º4, 0), (4, 0), (0, 3)

21. (3, 0), (2, 0), (º1, 0), (1, 4) 22. (0, 0), (º3, 0), (5, 0), (º2, 3)

$ %&&%&& $$ %'&PRACTICE AND APPLICATIONS

GEOMETRY CONNECTION

GUIDED PRACTICE


Number of sides, n 3 4 5 6 7 8

Number of diagonals, d 0 2 5 9 14 20


Skill Check ✓

STUDENT HELP

Extra Practiceto help you masterskills is on p. 949.

x 1 2 3 4 5 6

ƒ(x) º1 3 3 5 15 39

x 1 2 3 4 5 6

ƒ(x) 6 15 22 21 6 –29

x 1 2 3 4 5 6

ƒ(x) 0 8 12 12 8 0

x 1 2 3 4 5 6

ƒ(x) º1 º4 º3 8 35 84

� � � � ! " #


FINDING FINITE DIFFERENCES Show that the n th-order differences for the

given function of degree n are nonzero and constant.

23. ƒ(x) = x2 º 3x + 7 24. ƒ(x) = 2x3 º 5x2 º x 25. ƒ(x) = ºx3 + 3x2 º 2x º 3

26. ƒ(x) = x4 º 3x3 27. ƒ(x) = 2x4 º 20x 28. ƒ(x) = º4x2 + x + 6

29. ƒ(x) = ºx4 + 5x2 30. ƒ(x) = 3x3 º 5x2 º 2 31. ƒ(x) = º3x2 + 4x + 2

FINDING A MODEL Use finite differences and a system of equations to find

a polynomial function that fits the data. You may want to use a calculator.

32. 33.

34. 35.

36. 37.

38. 39.

40. 41.

42. 43.

44. PENTAGONAL NUMBERS The dot patterns show pentagonal numbers. A

formula for the nth pentagonal number is ƒ(n) = }

12

}n(3n º 1). Show that thisfunction has constant second-order differences.

45. HEXAGONAL NUMBERS A formula for the nth hexagonal number isƒ(n) = n(2n º 1). Show that this function has constant second-order differences.

46. SQUARE PYRAMIDAL NUMBERS The first six square pyramidal numbers areshown. Find a polynomial function that gives the nth square pyramidal number.

x 1 2 3 4 5 6

ƒ(x) º4 0 10 26 48 76

x 1 2 3 4 5 6

ƒ(x) 2 20 58 122 218 352

x 1 2 3 4 5 6

ƒ(x) º3 º8 º15 º21 º23 –18

x 1 2 3 4 5 6

ƒ(x) º5 0 9 16 15 0

x 1 2 3 4 5 6

ƒ(x) 20 º2 º4 2 4 º10

x 1 2 3 4 5 6

ƒ(x) 26 º4 º2 2 2 16

x 1 2 3 4 5 6

ƒ(x) 2 º5 º4 º1 º2 º13

x 1 2 3 4 5 6

ƒ(x) 0 6 2 6 12 º10

x 1 2 3 4 5 6

ƒ(x) º2 1 º4 º5 10 53

x 1 2 3 4 5 6

ƒ(x) –º2 º6 º6 4 30 78

x 1 2 3 4 5 6

ƒ(x) º4 º6 º2 14 48 106

x 1 2 3 4 5 6

ƒ(x) 17 28 33 32 25 12

ƒ(1) = 1 ƒ(2) = 5 ƒ(3) = 14 ƒ(4) = 30 ƒ(5) = 55 ƒ(6) = 91 ƒ(7) = 140

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 14–22Example 2: Exs. 23–31,

44, 45Example 3: Exs. 32–43,

46Example 4: Exs. 47–49

( ) * + , - . /FINDING MODELS In Exercises 47–49, use a graphing calculator to find a

polynomial model for the data.

47. GIRL SCOUTS The table shows the number of Girl Scouts (in thousands)from 1989 to 1996. Find a polynomial model for the data. Then estimate thenumber of Girl Scouts in 2000.

48. REAL ESTATE The table shows the average price (in thousands of dollars)of a house in the Northeastern United States for 1987 to 1995. Find a polynomialmodel for the data. Then predict the average price of a house in the Northeast in2000.

49. SPACE EXPLORATION The table shows the average speed y (in feet persecond) of a space shuttle for different times t (in seconds) after launch. Find a polynomial model for the data. When the space shuttle reaches a speed ofapproximately 4400 feet per second, its booster rockets fall off. Use the model to determine how long after launch this happens.

50. MULTI-STEP PROBLEM Your friend has a dog-walking service and your cousinhas a lawn-care service. You want to start a small business of your own. You aretrying to decide which of the two services you should choose. The profits for thefirst 6 months of the year are shown in the table.

a. Use finite differences to find a polynomial model for each business.

b. Writing You want to choose the business that will make the greater profit inDecember (when t = 12). Explain which business you should choose and why.

51. a. Substitute the expressions x, x + 1, x + 2, . . . , x + 5 for x in the function ƒ(x) = ax3 + bx2 + cx + d and show that third-order differences are constant.

b. The data below can be modeled by a cubic function. Set the variableexpressions you found in part (a) equal to the first-, second-, and third-orderdifferences for these values. Solve the equations to find the coefficients of thefunction that models the data. Check your work by substituting the originaldata values into the function.


t 1989 1990 1991 1992 1993 1994 1995 1996

y 231 253.3 273.8 284.1 294.1 303.6 368.6 383.7

x 1987 1988 1989 1990 1991 1992 1993 1994 1995

ƒ(x) 140 149 159.6 159 155.9 169 162.9 169 180

t 10 20 30 40 50 60 70 80

y 202.4 463.4 748.2 979.3 1186.3 1421.3 1795.4 2283.5

x 1 2 3 4 5 6

ƒ(x) º1 1 º3 º7 º5 9

Month, t 1 2 3 4 5 6

Profit, P 3 5 22 54 101 163

Month, t 1 2 3 4 5 6

Profit, P 3 21 41 68 107 163

TestPreparation

★★Challenge

EILEEN COLLINS

was selected byNASA for the astronautprogram in 1990. Since thenshe has become the firstwoman to pilot a spacecraftand the first woman tocommand a space shuttle.

RE

AL LIFE

RE

AL LIFE

FOCUS ON

PEOPLE

EXTRA CHALLENGE


DATA UPDATE of Statistical Abstract of the United States data at www.mcdougallittell.com INT

ERNET

Dog-walking

service

Lawn-care

service

( ) * + , - . /SOLVING QUADRATIC EQUATIONS Solve the equation. (Review 5.3 for 7.1)

52. 3x2 = 6 53. 16x2 = 4 54. 4x2 º 5 = 9

55. 6x2 + 3 = 16 56. ºx2 + 9 = 2x2 º 6 57. ºx2 + 2 = x2 + 1

SOLVING EQUATIONS Solve the equation by completing the square. (Review 5.5)

58. x2 + 12x + 27 = 0 59. x2 + 6x º 24 = 0 60. x2 º 3x º 18 = 0

61. 2x2 + 8x + 11 = 0 62. ºx2 + 14x + 15 = 0 63. 3x2 º 18x + 32 = 0

SUM OR DIFFERENCE OF CUBES Factor the polynomial. (Review 6.4)

64. 8x3 º 1 65. 27x3 + 8 66. 216x3 + 64 67. 8x3 º 125

68. 3x3 º 24 69. 8x3 + 216 70. 27x3 + 1000 71. 3x3 + 81

Find all the zeros of the polynomial function. (Lesson 6.7)

1. ƒ(x) = 2x3 º x2 º 22x º 15 2. ƒ(x) = x3 + 3x2 + 3x + 2

3. ƒ(x) = x4 º 3x3 º 2x2 º 6x º 8 4. ƒ(x) = 2x4 º x3 º 8x2 + x + 6

Write a polynomial of least degree that has real coefficients, the given zeros,

and a leading coefficient of 1. (Lesson 6.7)

5. º2, º2, 2 6. 0, 1, º3 7. 4, 2 + i, 2 º i

8. 2, 5, ºi 9. 4, 2 º 3i 10. 1 º i, 2 + 2i

Graph the function. Estimate the local maximums and minimums. (Lesson 6.8)

11. ƒ(x) = º(x º 2)(x + 3)(x + 1) 12. ƒ(x) = x(x º 1)(x + 1)(x + 2)

13. ƒ(x) = 2(x º 2)(x º 3)(x º 4) 14. ƒ(x) = (x + 1)(x + 3)2

Write a cubic function whose graph passes through the points. (Lesson 6.9)

15. (º2, 0), (2, 0), (º4, 0), (º1, 3) 16. (º1, 0), (4, 0), (2, 0), (º3, 1)

17. (3, 0), (0, 0), (5, 0), (2, 6) 18. (1, 0), (º3, 0), (º5, 0), (º4, 10)

Find a polynomial function that models the data. (Lesson 6.9)

19. 20.

21. SOCIAL SECURITY The table gives the number of children (in thousands)receiving Social Security for each year from 1988 to 1995. Use a graphing

calculator to find a polynomial model for the data. (Lesson 6.9)


MIXED REVIEW


x 1 2 3 4 5 6

ƒ(x) –º5 º6 º1 16 51 110

x 1 2 3 4 5 6

ƒ(x) –º1 º4 º3 8 35 84

Year 1988 1989 1990 1991 1992 1993 1994 1995

Number of children 3204 3165 3187 3268 3391 3527 3654 3734

( ) * + , - . /WHY did you learn it?

Use scientific notation to find the ratio of a state’spark space to its total area. (p. 328)

Estimate the amount of prize money awarded at atennis tournament. (p. 335)

Find maximum or minimum values of a function suchas oranges consumed in the U.S. (p. 377)

Write a polynomial model for the power needed to move a bicycle at a certain speed. (p. 342)

Find the dimensions of a block discovered by archeologists. (p. 347)

Find the dimensions of a sculpture. (p. 350)

Write a function for the average annual amount of money spent per person at the movies. (p. 358)

Find dimensions for a candle-wax model of the Louvre pyramid. (p. 361)

Write and use a polynomial model for the speed of a space shuttle. (p. 385)

Find the maximum volume and dimensions of a box made from a piece of cardboard. (p. 375)

Chapter SummaryCHAPTER

6

WHAT did you learn?

Use properties of exponents to evaluate and simplify expressions. (6.1)

Evaluate polynomial functions using direct or synthetic substitution. (6.2)

Sketch and analyze graphs of polynomial functions. (6.2, 6.8)

Add, subtract, and multiply polynomials. (6.3)

Factor polynomial expressions. (6.4)

Solve polynomial equations. (6.4)

Divide polynomials using long division or synthetic division. (6.5)

Find zeros of polynomial functions. (6.6, 6.7)

Use finite differences and cubic regression to find polynomial models for data. (6.9)

Use polynomials to solve real-life problems. (6.1–6.9)

How does Chapter 6 fit into the BIGGER PICTURE of algebra?

Chapter 6 contains the fundamental theorem of algebra. Finding the solutions of apolynomial equation is the most classic problem in all of algebra. It is equivalent tofinding the zeros of a polynomial function. Real-life situations have been modeled bypolynomial functions for hundreds of years.

How did you make and use a flow chart?

Here is a flow chart for finding allthe zeros of a polynomial function,following the Study Strategy onpage 322.

STUDY STRATEGY

Flow Chart

387

Graph the function. Approximate x-intercepts.

Factorable?

Use factortheorem.

Write all zeros.number of zeros = degree of function

Integercoefficients?

Use rationalzero theorem.

Use technologyto approximatezeros.

yes

yesno

no

Finding Zeros of a Polynomial

Write infactored form.

0 1 2 3 4 5 6 7


Chapter ReviewCHAPTER

6

• scientific notation, p. 325

• polynomial function, p. 329

• leading coefficient, p. 329

• constant term, p. 329

• degree of a polynomial function, p. 329

• standard form of a polynomial function, p. 329

• synthetic substitution, p. 330

• end behavior, p. 331

• factor by grouping, p. 346

• quadratic form, p. 346

• polynomial long division, p. 352

• remainder theorem, p. 353

• synthetic division, p. 353

• factor theorem, p. 354

• rational zero theorem, p. 359

• fundamental theorem of algebra, p. 366

• repeated solution, p. 366

• local maximum, p. 374

• local minimum, p. 374

• finite differences, p. 380

VOCABULARY

6.1 USING PROPERTIES OF EXPONENTS

You can use properties of exponents to evaluate numerical expressionsand to simplify algebraic expressions.

= = }24

93

}x10 º 10y5 º 6 = 27x0yº1 = }

2y7} all positive exponents

35x2 • 5y5

}

9x10y6

(3x2y)5

}

9x10y6


1. S}

23

}D2• (6xyº1)3 2. x4(xº5x3)2 3. 4. }

y

5º

x2

2} • }

25

1

x2y}

º63xy9

}

18xº2y3

6.2 EVALUATING AND GRAPHING POLYNOMIAL FUNCTIONS

Use direct or synthetic substitution to evaluate a polynomial function.

Evaluate ƒ(x) = x3 º 2x º 1 when x = 3 (synthetic substitution):

To graph, make a table of values, plot points, and identify end behavior.

The leading coefficient is positive and the degree is odd, so ƒ(x) ˘ º‡ as x ˘ º‡ and ƒ(x) ˘ +‡ as x ˘ +‡.

Examples onpp. 329–332

Use synthetic substitution to evaluate the polynomial function for the given value of x.

5. ƒ(x) = x3 + 3x2 º 12x + 7, x = 3 6. ƒ(x) = x4 º 5x3 º 3x2 + x º 5, x = º1

EXAMPLE

EXAMPLES


x º3 º2 º1 0 1 2 3

ƒ(x) º22 º5 0 º1 º2 3 20

8 99 :3 1 0 º2 º1

3 9 21

1 3 7 20 ¯ ƒ(3) = 20

0 1 2 3 4 5 6 7

Chapter Review 389

6.3 ADDING, SUBTRACTING, AND MULTIPLYING POLYNOMIALS

You can add, subtract, or multiply polynomials.

(x º 3)(x2 + 5x º 1) = (x º 3)(x2) + (x º 3)(5x) + (x º 3)(º1)

= x3 º 3x2 + 5x2 º 15x º x + 3

= x3 + 2x2 º 16x + 3

4x3 + 2x2 + 1º (x2 + x º 5)}}}}}}}}

4x3 + x2 º x + 6



10. (3x3 + x2 + 1) º (x3 + 3) 11. (x º 3)(x2 + x º 7) 12. (x + 3)(x º 5)(2x + 1)

6.4 FACTORING AND SOLVING POLYNOMIAL EQUATIONS

You can solve some polynomial equations by factoring.

Factor 8x3 º 125. Solve x3 º 3x2 º 5x + 15 = 0.

8x3 º 125 = (2x)3 º 53 x2(x º 3) º 5(x º 3) = 0

= (2x º 5)((2x)2 + (2x • 5) + 52) (x º 3)(x2 º 5) = 0

= (2x º 5)(4x2 + 10x + 25) x = 3 or x = ±Ï5w


Find the real-number solutions of the equation.

13. x3 + 64 = 0 14. x4 º 6x2 = 27 15. x3 + 3x2 º x º 3 = 0

EXAMPLES

EXAMPLES

6.5 THE REMAINDER AND FACTOR THEOREMS

You can use polynomial long division, and in some cases syntheticdivision, to divide polynomials.

x2 º 7x + 6

x + 9qx3w +w 2wx2w ºw 5w7wxw+w 5w4wx3 + 9x2}}}

º7x2 º 57xº7x2 º 63x}}}}

6x + 546x + 54}}}

0


Divide. Use synthetic division if possible.

16. (x4 + 5x3 º x2 º 3x º 1) ÷ (x º 1) 17. (2x3 º 5x2 + 5x + 4) ÷ (2x º 5)

EXAMPLES

Divide 3x3 + 2x2 º x + 4 by x + 5.

}

3x3 + 2x

x+

2 º5

x + 4} = 3x2 º 13x + 64 + }

ºx +

3156

}

º5 3 2 º1 4

º15 65 º320

3 º13 64 º316

Graph the polynomial function.

7. ƒ(x) = ºx3 + 2 8. ƒ(x) = x4 º 3 9. ƒ(x) = x3 º 4x + 1

= x2 º 7x + 6x3 + 2x2 º 57x + 54}}}

x + 9

; < = > ? @ A B


FINDING ZEROS OF POLYNOMIAL FUNCTIONSExamples on pp. 359–361

and pp. 366–368

Find all the real zeros of the function.

18. ƒ(x) = x3 + 12x2 + 21x + 10 19. ƒ(x) = x4 + x3 º x2 + x º 2

6.8

6.6–6.7

ANALYZING GRAPHS OF POLYNOMIAL FUNCTIONS

You can identify x-intercepts and turning points when you analyze the graph of a polynomial function.

The graph of ƒ(x) = 3x3 º 9x + 6 has

• two x-intercepts, º2 and 1.

• a local maximum at (º1, 12).

• a local minimum at (1, 0).


Graph the polynomial function. Identify the x-intercepts and the points where

the local maximums and local minimums occur.

20. ƒ(x) = (x º 2)2(x + 2) 21. ƒ(x) = x3 º 3x2 22. ƒ(x) = 3x4 + 4x3

You can use the rational zero theorem and the fundamental theorem ofalgebra to find all the zeros of a polynomial function.

ƒ(x) = x4 + 3x3 º 5x2 º 21x + 22 Possible rational zeros: }±1, ±2, ±

111, ±22}

Using synthetic division, you can find that the rational zeros are 1 and 2.The degree of ƒ is 4, so ƒ has 4 zeros. To find the other two zeros, write in factoredform: ƒ(x) = (x º 1)(x º 2)(x2 + 6x + 11). Solve x2 + 6x + 11 = 0: x = º3 ± Ï2wi.

So the zeros of ƒ(x) = x4 + 3x3 º 5x2 º 21x + 22 are 1, 2, º3 + Ï2wi, º3 º Ï2wi.

EXAMPLE

EXAMPLE

6.9 MODELING WITH POLYNOMIALS

Sometimes you can use finite differences or cubic regression to find a polynomial model for a set of data.

ƒ(1) ƒ(2) ƒ(3) ƒ(4) ƒ(5) ƒ(6)

º1 2 7 14 23 34 function values

3 5 7 9 11 first-order differences

2 2 2 2 second-order differences

Since second-order differences are nonzero and constant, the data set can be modeled by apolynomial function of degree 2. The function is ƒ(x) = x2 º 2.


23. Show that the third-order differences for the function ƒ(n) = n3 + 1 are nonzeroand constant.

24. Write a cubic function whose graph passes through points (1, 0), (º1, 0), (4, 0),and (2, º12). Use cubic regression on a graphing calculator to verify your answer.

EXAMPLE

8 9C :

; < = > ? @ A B

Chapter Test 391

Chapter TestCHAPTER

6


1. x7• }

x

12} 2. (32x6)3 3. }

x

xº

9

2} 4. (8x3y2)º3 5. }

1

6

5

x

x4y

2y5

} •

Describe the end behavior of the graph of the polynomial function. Then

evaluate the function for x = º4, º3, º2, . . ., 4. Then graph the function.

6. y = x4 º 2x2 º x º 1 7. y = º3x3 º 6x2 8. y = (x º 3)(x + 1)(x + 2)


9. (3x2 º 5x + 7) º (2x2 + 9x º 1) 10. (2x º 3)(5x2 º x + 6) 11. (x º 4)(x + 1)(x + 3)

Factor the polynomial.

12. 64x3 + 343 13. 400x2 º 25 14. x4 + 8x2 º 9 15. 2x3 º 3x2 + 4x º 6

Solve the equation.

16. 3x4 º 11x2 º 20 = 0 17. 81x4 = 16 18. 4x3 º 8x2 º x + 2 = 0

Divide. Use synthetic division if possible.

19. (8x4 + 5x3 + 4x2 º x + 7) ÷ (x + 1) 20. (12x3 + 31x2 º 17x º 6) ÷ (x + 3)

List all the possible rational zeros of ƒ using the rational zero theorem. Then

find all the zeros of the function.

21. ƒ(x) = x3 º 5x2 º 14x 22. ƒ(x) = x3 + 4x2 + 9x + 36 23. ƒ(x) = x4 + x3 º 2x2 + 4x º 24

Write a polynomial function of least degree that has real coefficients, the given

zeros, and a leading coefficient of 1.

24. 1, º3, 4 25. 2, 2, º1, 0 26. 5, 2i, º2i 27. 3, º3, 2 º i

28. Use technology to approximate the real zeros of ƒ(x) = 0.25x3 º 7x2 + 15.

29. Identify the x-intercepts, local maximum, and local minimum of the graph of

ƒ(x) = }

19

}(x º 3)2(x + 3)2. Then describe the end behavior of the graph.

30. Show that ƒ(x) = x4 º 2x + 8 has nonzero constant fourth-order differences.

31. The table gives the number of triangles that point upward that you can find in a large triangle that is n units on a side and divided into triangles that are eachone unit on a side. Find a polynomial model for ƒ(n).

32. CELLS An adult human body contains about 75,000,000,000,000 cells. Each is about 0.001 inch wide. If the cells were laid end to end to form a chain, about how long would the chain be in miles? Give your answer inscientific notation.

6x3y2

}

5xy

ƒ(2) 5 4

n 1 2 3 4 5 6 7

ƒ(n) 1 4 10 20 35 56 84

; < = > ? @ A B


Chapter Standardized TestCHAPTER

6

1. MULTIPLE CHOICE What is the value of º40?

¡A 4 ¡B 1 ¡C 0

¡D º1 ¡E º4

2. MULTIPLE CHOICE What is the value of ƒ(x) = 7x4 º 3x3 + 8x2 + x º 9 when x = º1?

¡A 8 ¡B 4 ¡C 2

¡D º8 ¡E º14

3. MULTIPLE CHOICE Which statement about the endbehavior of the graph of ƒ(x) = x4 + 1 is true?

¡A ƒ(x) ˘ +‡ as x ˘ º‡.

¡B ƒ(x) ˘ +‡ as x ˘ 0.

¡C ƒ(x) ˘ º‡ as x ˘ º‡.

¡D ƒ(x) ˘ º‡ as x ˘ 0.

¡E ƒ(x) ˘ º‡ as x ˘ +‡.

4. MULTIPLE CHOICE For 1992 through 1995, thenumber of grocery stores in the United States can bemodeled by G = 0.03t2 º 1.5t + 171, where G isthe number of stores in thousands and t is the numberof years since 1990. The average sales per grocerystore can be modeled by S = 4.7t2 + 49.1t + 2009,where S is sales in thousands of dollars. What werethe approximate total sales in millions of dollars forgrocery stores in the United States in 1994?

¡A 3.8 ª 10º1¡B 3.8 ª 101

¡C 3.8 ª 105¡D 3.8 ª 108

¡E 3.8 ª 1011

5. MULTIPLE CHOICE Which polynomial has thefactorization (2x + 1)(4x2 º 2x + 1)?

¡A 2x3 º 1 ¡B 8x3 º 1

¡C 2x3 + 1 ¡D 4x3 + 1

¡E 8x3 + 1

6. MULTIPLE CHOICE What are all the real solutionsof the equation x5 = 256x?

¡A 0, ±4 ¡B 4, º4 ¡C ±4, ±4i

¡D 0, ±4i ¡E 0, ±4, ±4i

7. MULTIPLE CHOICE What is the quotient of(4x3 º 11x2 º 9x º 5) ÷ (x º 4)?

¡A 4x3 + 5x2 + 11x + 39

¡B 4x2 + 5x + 11 + }x

3º9

4}

¡C 4x2 + 5x + 11 +

¡D 4x2 º 27x + 99 º }x4º01

4}

¡E 4x2 º 27x + 99 º }x4+01

4}

8. MULTIPLE CHOICE What are all the rational zerosof ƒ(x) = x3 º 8x2 + x + 42?

¡A º2, º3, º7 ¡B 2, 3, 7

¡C 2, º3, º7 ¡D 0, 6, 7

¡E º2, 3, 7

9. MULTIPLE CHOICE How many zeros does thefunction ƒ(x) = º3x4 + x + 2 have?

¡A 0 ¡B 1 ¡C 2

¡D 3 ¡E 4

10. MULTIPLE CHOICE Which function is graphed?

¡A ƒ(x) = (x + 3)(x º 1)(x º 4)

¡B ƒ(x) = 7(x + 3)(x º 1)(x º 4)

¡C ƒ(x) = }

172}(x º 3)(x + 1)(x + 4)

¡D ƒ(x) = }

172}(x + 3)(x º 1)(x º 4)

¡E ƒ(x) = º}

172}(x + 3)(x º 1)(x º 4)

D EF G(0, 7)

(4, 0)

(1, 0)

(23, 0)

39}}}

4x3 º 11x2 º 9x º 5

TEST-TAKING STRATEGY The mathematical portion of the SAT is based on concepts and skills taught in high

school mathematics courses. The best way to prepare for the SAT is to keep up with your day-to-day studies.

H I J K L M N O

Chapter Standardized Test 393

QUANTITATIVE COMPARISON In Exercises 11 and 12, choose the statement

that is true about the given quantities.

¡A The quantity in column A is greater.

¡B The quantity in column B is greater.

¡C The two quantities are equal.

¡D The relationship cannot be determined from the given information.

11.

12.

13. MULTI-STEP PROBLEM You are designing a monument for the city park. The monument is to be a rectangular prism with dimensions x + 1 feet, x º 5 feet, and x º 6 feet.

a. Write a function ƒ(x) for the volume of the monument.

b. Use a graphing calculator to graph ƒ(x) for º10 ≤ x ≤ 20.

c. Writing Look back at your graph from part (b). Identify the localmaximums and local minimums. Do these values represent maximum and minimum possible volumes of the monument? Explain.

d. If the volume of the monument is to be 220 cubic feet, what will thedimensions be?

14. MULTI-STEP PROBLEM The numbers in the table give the volumes of the first six prisms in a sequence.

a. Use finite differences to determine the degree of ƒ.

b. Use a system of equations to find a polynomial model for ƒ(n) in standard form.

c. Writing Factor the polynomial. Explain how the factors are related to thedimensions of the prism.

d. Use your model to find the volume of the 50th prism in the sequence.

e. Sketch a graph of your model and label the points that represent the first six prisms. What is the domain of the function?

Column A Column B

xº2 x2

Degree of ƒ(x) = x4 º 7x + 13 Degree of ƒ(x) = 4x3 + 2x2 º x + 1

Prism (n) 0 1 2 3 4 5r

Volume, ƒ(n) 6 24 60 120 210 336

ƒ(2) 5 60ƒ(1) 5 24ƒ(0) 5 6 ƒ(3) 5 120

H I J K L M N O

394 Chapters 1–6

Cumulative Practice for Chapters 1–6

CHAPTER

6

Solve the equation. (1.3, 1.7)

1. 5x + 4 = º21 2. 3(2x + 5) = 69 3. |x º 2| = 6 4. |7 º 3x| = 23

Solve the inequality. Then graph your solution. (1.6, 1.7)

5. 10 º 4x > º2 6. 0 ≤ 2x º 8 ≤ 14 7. |x º 3| < 5 8. |5x + 2| ≥ 17

Find the slope of the line passing through the given points. (2.2)

9. (4, 1), (º2, 1) 10. (º3, 0), (0, 2) 11. (º1, º5), (2, 7) 12. (º4, 4), (1, º3)

Graph the equation or inequality. (2.3, 2.6–2.8)

13. y = º2x º 1 14. 3x º 7y = 21 15. x = º4

16. y > }

32}x + 2 17. 2x + 6y ≤ 12 18. y = |x| + 3

19. y = º2|x + 4| º 1 20. ƒ(x) = 21. ƒ(x) =

Write an equation of the line with the given characteristics. (2.4)

22. slope: 3, y-intercept: º2 23. points on line: (º1, 9), (1, 1) 24. vertical line through (º8, 6)

Solve the system of linear equations using any method. (3.1, 3.2, 3.6, 4.3, 4.5)

25. x + y = 8 26. 3x º 4y = 5 27. x + y + z = 4 28. x º y + z = 12x º y = 1 2x + 2y = 1 x º 4y + 3z = 10 ºx + y + 2z = 2

º4x + y + z = º1 x + y + z = º3

Graph the ordered triple or equation in a three-dimensional coordinate system. (3.5)

29. (º1, º3, 0) 30. (2, 4, º2) 31. 4x + 2y + z = 4 32. 5x + 5y + 2z = 10

Perform the indicated operation. (4.1, 4.2)

33. F G + F G 34. º6 F G 35. F GF G

Evaluate the determinant of the matrix. (4.3)

36. F G 37. F G 38. F G 39. F GFind the inverse of the matrix. (4.4)

40. F G 41. F G 42. F G 43. F GGraph the equation or inequality. (5.1, 5.7, 6.2, 6.8)

44. y = x2 + 8x + 16 45. y = º(x º 1)2 + 3 46. y = 2(x + 1)(x º 3)

47. y ≤ }

14}x2 º 3 48. y < º2x2 + 4x + 5 49. y = x3 º 4x2 + x + 7

50. y = º3x4 + 9x2 º 2 51. y = º(x + 2)(x º 1)(x º 2) 52. y = 2x2(x º 3)2

º21

4º2

94

42

º27

º14

23

º5º7

1º2

0

071

º138

128

914

3º5º2

16

0º3

2º2

º54

87

º21

2º3

º53

16

3º2º1

2º4º5

10

º8º3

7º2

º34

ºx, if x < 1x º 2, if x ≥ 1

º2, if x ≤ 03, if x > 0

H I J K L M N OSolve the equation or inequality. (5.2–5.7, 6.4)

53. 3x2 º 7 = 2(x2 + 3) 54. 4x2 + 12x + 9 = 0 55. x2 + 64 = 0

56. x2 + 4x = 4 57. 100 º x2 ≥ 0 58. x2 º 6 > º5x

59. x4 º 5x2 + 4 = 0 60. 3x4 º 15x3 = 0 61. 2x3 + 4x2 º 3x º 6 = 0

Write the expression as a complex number in standard form. (5.4)

62. }

74

+º

3i

i} 63. 4i(5 º 8i) 64. (9 + 5i)(9 º 5i) 65. (6 º 2i) º (º3 º 4i)

Write a quadratic function in the specified form whose graph has the given

characteristics. (5.8)

66. vertex form 67. intercept form 68. standard formvertex: (5, 3) x-intercepts: º3, º2 points on graph:point on graph: (7, 11) point on graph: (0, º6) (1, 4), (3, º4), (6, º61)

Simplify the expression. (6.1)

69. (6xy3)2 70. 7xº10y4 71. S}

54}Dº2

72. }

3x

2

2y

x

º1

} • }

1

3

0

y

x

º

2

3

y}

Perform the indicated operation. (6.3, 6.5)

73. (x º 3)(x3 º 2x2 + 5x º 12) 74. (7x3 º 9x + 2) + (5x3 + 9x) 75. (x4 º 3x3 + 8x2 º 2) ÷ (x + 2)

Find all the zeros of the function. (6.6, 6.7)

76. ƒ(x) = 2x3 º 5x2 º 4x + 3 77. ƒ(x) = x4 º 25 78. ƒ(x) = x3 + 11x2 + x + 11

Write a cubic function whose graph passes through the given points. (6.9)

79. (º4, 0), (º1, 0), (1, 0), (º2, 6) 80. (º6, 0), (0, 0), (3, 0), (6, º144)

81. SIMPLE INTEREST The formula for simple interest is I = Prt. Solve theformula for r. Then find the annual interest rate if a $1000 deposit earns $165 ofsimple interest in 3 years. (1.4)

82. COST OF BREAD The table gives the number of one-pound loaves of breadyou could buy for $1.00 in the United States for various years since 1900. Makea scatter plot of the data and describe the correlation shown. (2.5)

DATA UPDATE of Bureau of Labor Statistics data at www.mcdougallittell.com

83. PHONE RATES A long distance carrier charges a flat rate of $.09 perminute for telephone calls. A second carrier charges $.30 for the first minute and$.06 for each additional minute. After how many minutes will the second carrierbe less expensive than the first carrier? (3.2)

84. CRYPTOGRAMS Use the matrix A = F G and the coding

information on page 225 to encode the message EXIT NOW. (4.4)

85. Pluto is about 3,660,000,000 mi from the sun. Lighttravels through space at a speed of about 671,000,000 mi/h. Use scientificnotation to find how long it takes light from the sun to reach Pluto. (6.1)

SCIENCE CONNECTION

58

º21

INT

ERNET

Cumulative Practice 395

Years since 1900, t 13 30 50 70 90 97

Loaves of bread, b 17.8 11.6 6.9 4.1 1.4 1.1

H I J K L M N O

396 Chapters 4–6

PROJECTApplying Chapters

4–6Magic Squares

OBJECTIVE Explore the mathematics behind magic squares.

Materials: paper, pencil

A magic square is a square array of consecutive integers, usually (but notalways) beginning with 1, for which the sum of the entries in each row,column, and diagonal is the same. This common sum is called the magicconstant.

For example, a 4 3 4 magic square with a magic constant of 34 is shown. This square appears in the engraving Melancholia, which was created in 1514 by the German artist and mathematician Albrecht Dürer.

Magic squares were discovered in China around 2200 B.C. and later spread toIndia, Japan, and eventually to Europe. The challenge of creating magicsquares has fascinated mathematicians and puzzle lovers for many centuries.

HOW TO MAKE A 3 3 3 MAGIC SQUARE

Draw a 3 ª 3 square. You want to

use the integers 1 through 9 to fill

in the square. Start by writing the

middle value, 5, in the center.

INVESTIGATION

1. Think of a magic square as a matrix. Suppose a 3 3 3 matrix containing all 2’s isadded to the magic square in Step 2. Is the resulting matrix also a magic square?If so, what is the magic constant?

2. Generalize your work from Exercise 1 by adding a 3 3 3 matrix containing alla’s, where a represents any integer, to the magic square in Step 2. Is the resultalways a magic square? If so, what is the magic constant in terms of a?

3. Use the integers 1 through 9 to make another 3 3 3 magic square. Add yoursquare to the one in Step 2. Is the result a magic square? Explain. (Rememberthat the square’s rows, columns, and diagonals must have the same sum and thenumbers in the square must be consecutive integers.)

4. Use scalar multiplication to multiply the magic square in Step 2 by the scalar 2.Is the result a magic square? Explain.

1 Continue filling in numbers until

you have an arrangement where

the entries in each row, column,

and diagonal add up to 15.

2

5

4

3

8

9

5

1

2

7

6

P Q R S T U V WINVESTIGATION (continued)

5. The transpose of a matrix A is a matrix AT obtained by interchanging the rows and columns of A—the first row of A becomes the first column of AT, the second row of A becomes the second column of AT, and so on. Find thetranspose of the magic square in Step 2. Is the transpose also a magic square?

6. Copy and complete the 4 3 4 magic square shown. What reasoning did you useto place the remaining numbers?

7. The sum S of the first k positive integers is given by the quadratic function

S = }12

}k2 + }12

}k. Use this function to find the sum of the entries in the 3 ª 3 and

4 ª 4 magic squares from Step 2 and Exercise 6. Check your answers bycomputing the sums directly.

8. Consider an n 3 n magic square that contains the integers 1 through n2. Use thefunction from Exercise 7 to write a formula for the sum S of the entries in thesquare in terms of n. What type of function is this formula?

9. For an n 3 n magic square that contains the integers 1 through n2, write aformula for the square’s magic constant M in terms of n. (Hint: Note that themagic constant is the sum of all the entries in the square divided by the numberof rows or columns.) What type of function is this formula?

PRESENT YOUR RESULTS

Write a report to present your results.

• Include the 3 ª 3 and 4 ª 4 magic squares you made.

• Tell whether a magic square is produced by performing each of the following operations on an n ª n magic square A: adding the same integer to each entry of A,multiplying each entry of A by the same integer, adding another n ª n magic square to A, and taking the transpose of A.

• Include the formulas you found for the sum of the entries and for the magic constant of an n ª n magic square containing the integers 1through n2.

• Describe how you used your knowledge of matrices, quadratic functions, andhigher-degree polynomial functions in this project.

EXTENSION

Consider an n 3 n magic square containing the integers athrough a + n2 º 1. Such a magic square is shown at theright for n = 3 and a = 5. For this type of magic square,write formulas for the sum S of the entries and for themagic constant M in terms of n and a. Verify that yourformulas work for the magic square shown.

Project 397

8

7

12

13

9

5

6

11

10

7

9

13

3

8

10

14

4

polynomials and polynomial...

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