polynomial and rational functions 2 - weebly

Quadratic functions are often usedto model real-life phenomena, suchas the path of a diver.

127

SELECTED APPLICATIONSPolynomial and rational functions have many real-life applications. The applications listed below represent a small sample of the applications in this chapter.

• Path of a Diver,Exercise 77, page 136

• Data Analysis: Home Prices,Exercises 93–96, page 151

• Data Analysis: Cable Television,Exercise 74, page 161

• Advertising Cost,Exercise 105, page 181

• Athletics,Exercise 109, page 182

• Recycling,Exercise 112, page 195

• Average Speed,Exercise 79, page 196

• Height of a Projectile,Exercise 67, page 205

2.1 Quadratic Functions and Models

2.2 Polynomial Functions of Higher Degree

2.3 Polynomial and Synthetic Division

2.4 Complex Numbers

2.5 Zeros of Polynomial Functions

2.6 Rational Functions

2.7 Nonlinear Inequalities

Polynomial andRational Functions 22

© M

arti

n R

ose

/Bo

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arts

/Get

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ages

332521_0200.qxd 2/8/07 2:54 PM 27

The Graph of a Quadratic FunctionIn this and the next section, you will study the graphs of polynomial functions.In Section 1.6, you were introduced to the following basic functions.

Linear function

Constant function

Squaring function

These functions are examples of polynomial functions.

Polynomial functions are classified by degree. For instance, a constant func-tion has degree 0 and a linear function has degree 1. In this section, you will studysecond-degree polynomial functions, which are called quadratic functions.

For instance, each of the following functions is a quadratic function.

Note that the squaring function is a simple quadratic function that has degree 2.

The graph of a quadratic function is a special type of “U”-shaped curve called aparabola. Parabolas occur in many real-life applications—especially thoseinvolving reflective properties of satellite dishes and flashlight reflectors. Youwill study these properties in Section 10.2.

mx x 2x 1

kx 3x2 4

hx 9 14 x2

gx 2x 12 3

f x x2 6x 2

f x x2

f x c

f x ax b

128 Chapter 2 Polynomial and Rational Functions

What you should learn• Analyze graphs of quadratic

functions.

• Write quadratic functions instandard form and use theresults to sketch graphs offunctions.

• Use quadratic functions tomodel and solve real-life problems.

Why you should learn itQuadratic functions can be used to model data to analyzeconsumer behavior. For instance,in Exercise 83 on page 137, youwill use a quadratic functionto model the revenue earnedfrom manufacturing handheldvideo games.

Quadratic Functions and Models

© John Henley/Corbis

2.1

Definition of Polynomial FunctionLet be a nonnegative integer and let be realnumbers with The function given by

is called a polynomial function of x with degree n.

f x anxn an1xn1 . . . a2x2 a1x a0

an 0.an1, . . . , a2, a1, a0an,n

Definition of Quadratic FunctionLet and be real numbers with The function given by

Quadratic function

is called a quadratic function.

f x ax2 bx c

a 0.ca, b,

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All parabolas are symmetric with respect to a line called the axis ofsymmetry, or simply the axis of the parabola. The point where the axis intersectsthe parabola is the vertex of the parabola, as shown in Figure 2.1. If the leadingcoefficient is positive, the graph of

is a parabola that opens upward. If the leading coefficient is negative, the graph of

is a parabola that opens downward.

Leading coefficient is positive. Leading coefficient is negative.FIGURE 2.1

The simplest type of quadratic function is

Its graph is a parabola whose vertex is (0, 0). If the vertex is the point withthe minimum -value on the graph, and if the vertex is the point with themaximum -value on the graph, as shown in Figure 2.2.

Leading coefficient is positive. Leading coefficient is negative.FIGURE 2.2

When sketching the graph of it is helpful to use the graph ofas a reference, as discussed in Section 1.7.y x 2

f x ax2,

x

f x ax a( ) = , < 02

Maximum: (0, 0)

−3 −2 −1 1 2 3

−3

−2

−1

1

2

3

y

x

f x ax a( ) = , > 02

Minimum: (0, 0)

−3 −2 −1 1 2 3

−3

−2

−1

1

2

3

y

ya < 0,y

a > 0,

f x ax2.

x

Opens downward

Vertex ishighest point

Axis

f x ax bx c, a <( ) = + + 02

y

Opens upward

Vertex islowest point

Axis

xf x ax bx c, a( ) = + + 02 >

y

f x ax2 bx c

f x ax2 bx c

Section 2.1 Quadratic Functions and Models 129

Graph for 0.5, 1, and 2. How does

changing the value of affectthe graph?

Graph for 2, and 4. How does chang-

ing the value of affect thegraph?

Graph for 2, and 4. How does chang-

ing the value of affect thegraph?

k2,

k 4,y x2 k

h2,

h 4,y x h2

a0.5,

a 2, 1,y ax2

Exploration

332521_0201.qxd 2/8/07 2:56 PM Page 129

Sketching Graphs of Quadratic Functions

a. Compare the graphs of and

b. Compare the graphs of and

Solutiona. Compared with each output of “shrinks” by a factor of

creating the broader parabola shown in Figure 2.3.

b. Compared with each output of “stretches” by a factorof 2, creating the narrower parabola shown in Figure 2.4.

FIGURE 2.3 FIGURE 2.4

Now try Exercise 9.

In Example 1, note that the coefficient determines how widely the parabolagiven by opens. If is small, the parabola opens more widely thanif is large.

Recall from Section 1.7 that the graphs of and are rigid transformations of the graph of

For instance, in Figure 2.5, notice how the graph of can be transformedto produce the graphs of and

Reflection in x-axis followed by Left shift of two units followed by an upward shift of one unit a downward shift of three unitsFIGURE 2.5

x

y = x2

g(x) = (x + 2)2 − 3

−4 −3 −1 1 2

3

2

1

−2

−3(−2, −3)

y

x−2 2

−1

−2

2

y x= 2

(0, 1)

y

f(x) = −x2 + 1

gx x 22 3.f x x2 1y x2

y f x.y f xy f x,y f x ± c,y f x ± c,

aaf x ax2

a

x−2 −1 1 2

1

2

3

4

y x= 2

g x x( ) = 2 2y

x−2 −1 1 2

1

2

3

4

y x= 2

f x x( ) = 13

2

y

gx 2x2y x2,

13,f x

13x2y x2,

gx 2x2.y x2

f x 13x2.y x2


Example 1

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The Standard Form of a Quadratic FunctionThe standard form of a quadratic function is This formis especially convenient for sketching a parabola because it identifies the vertexof the parabola as

To graph a parabola, it is helpful to begin by writing the quadratic functionin standard form using the process of completing the square, as illustrated inExample 2. In this example, notice that when completing the square, you add andsubtract the square of half the coefficient of within the parentheses instead ofadding the value to each side of the equation as is done in Appendix A.5.

Graphing a Parabola in Standard Form

Sketch the graph of and identify the vertex and the axis ofthe parabola.

SolutionBegin by writing the quadratic function in standard form. Notice that the firststep in completing the square is to factor out any coefficient of that is not 1.

Write original function.

Factor 2 out of -terms.

Add and subtract 4 within parentheses.

After adding and subtracting 4 within the parentheses, you must now regroup theterms to form a perfect square trinomial. The can be removed from inside theparentheses; however, because of the 2 outside of the parentheses, you must mul-tiply by 2, as shown below.

Regroup terms.

Simplify.

Write in standard form.

From this form, you can see that the graph of is a parabola that opensupward and has its vertex at This corresponds to a left shift of twounits and a downward shift of one unit relative to the graph of as shownin Figure 2.6. In the figure, you can see that the axis of the parabola is the verticalline through the vertex,

Now try Exercise 13.

x 2.

y 2x2,2, 1.

f

2x 22 1

2x2 4x 4 8 7

f x 2x2 4x 4 24 7

4

4

422

2x2 4x 4 4 7

x 2x2 4x 7

f x 2x2 8x 7

x2

f x 2x2 8x 7

x

h, k.

f x ax h2 k.


The standard form of a quadraticfunction identifies four basictransformations of the graph of

a. The factor produces avertical stretch or shrink.

b. If the graph is reflectedin the -axis.

c. The factor representsa horizontal shift of units.

d. The term represents avertical shift of units.k

k

hx h2

xa < 0,

ay x2.

x−3 −1 1

1

2

3

4

y x= 2 2

f x x( ) = 2( + 2) 12 −

( 2, 1)− − x = 2−

y

FIGURE 2.6

Standard Form of a Quadratic FunctionThe quadratic function given by

is in standard form. The graph of is a parabola whose axis is the verticalline and whose vertex is the point If the parabola opensupward, and if the parabola opens downward.a < 0,

a > 0,h, k.x hf

a 0f x ax h2 k,

Example 2

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To find the -intercepts of the graph of you mustsolve the equation If does not factor, you canuse the Quadratic Formula to find the -intercepts. Remember, however, that aparabola may not have -intercepts.

Finding the Vertex and x-Intercepts of a Parabola

Sketch the graph of and identify the vertex and -intercepts.

SolutionWrite original function.

Factor out of -terms.

Regroup terms.


From this form, you can see that is a parabola that opens downward with vertexThe -intercepts of the graph are determined as follows.

Factor out

Factor.

Set 1st factor equal to 0.

Set 2nd factor equal to 0.

So, the -intercepts are and as shown in Figure 2.7.


Writing the Equation of a Parabola

Write the standard form of the equation of the parabola whose vertex is andthat passes through the point as shown in Figure 2.8.

SolutionBecause the vertex of the parabola is at the equation has the form

Substitute for and in standard form.

Because the parabola passes through the point it follows that So,

Substitute 0 for solve for

which implies that the equation in standard form is


f x 2x 12 2.

a.x;a 20 a0 12 2

f 0 0.0, 0,

khf x ax 12 2.

h, k 1, 2,

0, 0,1, 2

4, 0,2, 0x

x 4 x 4 0

x 2 x 2 0

x 2x 4 0

1. x2 6x 8 0

x3, 1.f

x 32 1

x2 6x 9 9 8

622

Add and subtract 9 withinparentheses.

x2 6x 9 9 8

x1 x2 6x 8

f x x2 6x 8

xf x x2 6x 8

xx

ax2 bx cax2 bx c 0.f x ax2 bx c,x


x−1 1 3 5

−3

−2

−4

−1

1

2(3, 1)

(2, 0) (4, 0)

y

f(x) = − (x − 3)2 + 1

y = −x2

FIGURE 2.7

x

y = f(x)(1, 2)

(0, 0)

1

1

2

y

FIGURE 2.8

Example 3

Example 4

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ApplicationsMany applications involve finding the maximum or minimum value of aquadratic function. You can find the maximum or minimum value of a quadraticfunction by locating the vertex of the graph of the function.

The Maximum Height of a Baseball

A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet persecond and at an angle of with respect to the ground. The path of the baseballis given by the function where is the height ofthe baseball (in feet) and is the horizontal distance from home plate (in feet).What is the maximum height reached by the baseball?

SolutionFrom the given function, you can see that and Because thefunction has a maximum when you can conclude that the baseballreaches its maximum height when it is feet from home plate, where is

feet.

At this distance, the maximum height is feet. The path of the baseball is shown in Figure 2.9.


Minimizing Cost

A small local soft-drink manufacturer has daily production costs ofwhere is the total cost (in dollars) and is the

number of units produced. How many units should be produced each day to yielda minimum cost?

SolutionUse the fact that the function has a minimum when From the givenfunction you can see that and So, producing

units

each day will yield a minimum cost.


x b

2a

1202(0.075 800

b 120.a 0.075x b2a.

xCC 70,000 120x 0.075x2,

156.25 3 81.125 f 156.25 0.0032156.252

x b

2a

120.0032 156.25 x

b2a

xxx b2a,

b 1.a 0.0032

xf xf x 0.0032x2 x 3,

45


Vertex of a Parabola

The vertex of the graph of is

1. If has a minimum at

2. If has a maximum at x b2a

.a < 0,

x b

2a.a > 0,

b

2a, f

b2a.f x ax2 bx c

Distance (in feet)

Hei

ght (

in f

eet)

100 200 300

20

40

60

80

100

x

y

(156.25, 81.125)

Baseball

f(x) = −0.0032x2 + x + 3

FIGURE 2.9

Example 5

Example 6

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In Exercises 1– 8, match the quadratic function with itsgraph. [The graphs are labeled (a), (b), (c), (d), (e), (f ), (g),and (h).]

(a) (b)

(c) (d)

(e) (f)

(g) (h)

1. 2.

3. 4.

5. 6.

7. 8.

In Exercises 9–12, graph each function. Compare the graphof each function with the graph of

9. (a) (b)

(c) (d)

10. (a) (b)

(c) (d)

11. (a) (b)

(c) (d)

12. (a)

(b)

(c)

(d)

In Exercises 13–28, sketch the graph of the quadratic func-tion without using a graphing utility. Identify the vertex,axis of symmetry, and -intercept(s).

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25.

26.

27.

28. f x 13x2 3x 6

f x 14x2 2x 12

f x 2x2 x 1

hx 4x2 4x 21

f x x2 4x 1f x x2 2x 5

f x x2 3x 14f x x2 x

54

gx x2 2x 1hx x2 8x 16

f x x 62 3f x x 52 6

f x 16 14 x2f x

12x2 4

hx 25 x2f x x2 5

x

kx 2x 12 4

hx 12x 22 1

gx 12x 12

3

f x 12x 22 1

kx x 32hx 13 x2

3

gx 3x2 1f x x 12

kx x2 3hx x2 3

gx x2 1f x x2 1

kx 3x2hx 32 x2

gx 18 x2f x

12 x2

y x2.

f x x 42f x x 32 2

f x x 12 2f x 4 (x 2)2

f x 3 x2f x x2 2

f x x 42f x x 22

x

(0, 3)

−24−4

−4

4

y

x

(2, 0)

2−2 4 6

2

6

4

y

x

(2, 4)

2−2 6

2

4

y

x

(3, 2)−2 4

−2

2

−4

−6

6

y

x(4, 0)

2 4 6 8−2

−4

−6

y

x

(− 4, 0)

−2−2

−6 − 4

2

4

6

y

x

(0, −2)2−2 4−4

2

4

6

y

x

(−1, −2)2−4

2

4

6

y


Exercises 2.1

VOCABULARY CHECK: Fill in the blanks.

1. A polynomial function of degree and leading coefficient is a function of the form where is a ________ ________ and are ________ numbers.

2. A ________ function is a second-degree polynomial function, and its graph is called a ________.

3. The graph of a quadratic function is symmetric about its ________.

4. If the graph of a quadratic function opens upward, then its leading coefficient is ________ and the vertex of the graph is a ________.

5. If the graph of a quadratic function opens downward, then its leading coefficient is ________ and the vertex of the graph is a ________.

a1nan 0f x anxn an1xn1 . . . a1x a0

ann

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In Exercises 29–36, use a graphing utility to graph thequadratic function. Identify the vertex, axis of symmetry,and -intercepts. Then check your results algebraically bywriting the quadratic function in standard form.

29. 30.

31. 32.

33. 34.

35. 36.

In Exercises 37– 42, find the standard form of the quadraticfunction.

37. 38.

39. 40.

41. 42.

In Exercises 43–52, write the standard form of the equationof the parabola that has the indicated vertex and whosegraph passes through the given point.

43. Vertex: point:

44. Vertex: point:

45. Vertex: point:

46. Vertex: point:

47. Vertex: point:

48. Vertex: point:

49. Vertex: point:

50. Vertex: point:

51. Vertex: point:

52. Vertex: point:

Graphical Reasoning In Exercises 53–56, determine the -intercept(s) of the graph visually. Then find the

-intercepts algebraically to confirm your results.

53. 54.

55. 56.

In Exercises 57–64, use a graphing utility to graph thequadratic function. Find the -intercepts of the graph andcompare them with the solutions of the correspondingquadratic equation when

57.

58.

59.

60.

61.

62.

63.

64.

In Exercises 65–70, find two quadratic functions, one thatopens upward and one that opens downward, whosegraphs have the given -intercepts. (There are manycorrect answers.)

65. 66.

67. 68.

69. 70. 52, 0, 2, 03, 0, 1

2, 04, 0, 8, 00, 0, 10, 05, 0, 5, 01, 0, 3, 0

x

f x 7

10x2 12x 45f x

12x2 6x 7

f x 4x2 25x 21

f x 2x2 7x 30

f x x2 8x 20

f x x2 9x 18

f x 2x2 10x

f x x2 4x

f x 0.

x

x−4

−4

−6 2

2

−2

y

x

−4

−4

−8

8

y

y 2x2 5x 3y x2 4x 5

x2 4 6

2

4

6

8

y

8−8

−4

x

y

y x2 6x 9y x2 16

xx

6110, 326, 6;7

2, 163 5

2, 0;2, 45

2, 34;

2, 014, 32;

1, 02, 2;7, 155, 12;

0, 22, 3;1, 23, 4;

2, 34, 1;0, 92, 5;

x

(2, 0)

(3, 2)

−2 42 6

2

4

6

8

y

x(−3, 0)

(−1, 0)

(−2, 2)

−4−6

−6

2

2

y

x

(−2, −1)

(0, 3)

−4−6 2

2

6

y

x

(−1, 4)

(−3, 0)

(1, 0)

−2

−2

−4

−4

2

2

y

x

y

2 4−2

−6

−4

2(1, 0)

(0, 1)(−1, 0)

x(0, 1)

(1, 0)

−2 2 4

6

8

y

f x 35x2 6x 5gx

12x2 4x 2

f x 4x2 24x 41f x 2x2 16x 31

f x x2 10x 14gx x2 8x 11

f x x2 x 30f x x2 2x 3

x

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In Exercises 71–74, find two positive real numbers whoseproduct is a maximum.

71. The sum is 110.

72. The sum is

73. The sum of the first and twice the second is 24.

74. The sum of the first and three times the second is 42.

75. Numerical, Graphical, and Analytical Analysis Arancher has 200 feet of fencing to enclose two adjacent rectangular corrals (see figure).

(a) Write the area of the corral as a function of

(b) Create a table showing possible values of and thecorresponding areas of the corral. Use the table to esti-mate the dimensions that will produce the maximumenclosed area.

(c) Use a graphing utility to graph the area function. Usethe graph to approximate the dimensions that will pro-duce the maximum enclosed area.

(d) Write the area function in standard form to findanalytically the dimensions that will produce the max-imum area.

(e) Compare your results from parts (b), (c), and (d).

76. Geometry An indoor physical fitness room consists of a rectangular region with a semicircle on each end (seefigure). The perimeter of the room is to be a 200-metersingle-lane running track.

(a) Determine the radius of the semicircular ends of theroom. Determine the distance, in terms of around theinside edge of the two semicircular parts of the track.

(b) Use the result of part (a) to write an equation, in termsof and for the distance traveled in one lap aroundthe track. Solve for

(c) Use the result of part (b) to write the area ofthe rectangular region as a function of What dimen-sions will produce a maximum area of the rectangle?

77. Path of a Diver The path of a diver is given by

where is the height (in feet) and is the horizontaldistance from the end of the diving board (in feet). What isthe maximum height of the diver?

78. Height of a Ball The height (in feet) of a punted foot-ball is given by

where is the horizontal distance (in feet) from the point atwhich the ball is punted (see figure).

(a) How high is the ball when it is punted?

(b) What is the maximum height of the punt?

(c) How long is the punt?

79. Minimum Cost A manufacturer of lighting fixtures hasdaily production costs of

where is the total cost (in dollars) and is the number ofunits produced. How many fixtures should be producedeach day to yield a minimum cost?

80. Minimum Cost A textile manufacturer has daily produc-tion costs of

where is the total cost (in dollars) and is the number ofunits produced. How many units should be produced eachday to yield a minimum cost?

81. Maximum Profit The profit (in dollars) for a companythat produces antivirus and system utilities software is

where is the number of units sold. What sales level willyield a maximum profit?

x

P 0.0002x2 140x 250,000

P

xC

C 100,000 110x 0.045x 2

xC

C 800 10x 0.25x2

xNot drawn to scale

y

x

y 16

2025x2

95

x 1.5

y

xy

y 4

9x2

24

9x 12

x.A

y.y,x

y,

y

x

x

x.A

xx

y

S.


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82. Maximum Profit The profit (in hundreds of dollars)that a company makes depends on the amount (inhundreds of dollars) the company spends on advertisingaccording to the model

What expenditure for advertising will yield a maximumprofit?

83. Maximum Revenue The total revenue earned (inthousands of dollars) from manufacturing handheld videogames is given by

where is the price per unit (in dollars).

(a) Find the revenue earned for each price per unit givenbelow.

$20

$25

$30

(b) Find the unit price that will yield a maximum revenue.What is the maximum revenue? Explain your results.

84. Maximum Revenue The total revenue earned per day(in dollars) from a pet-sitting service is given by

where is the price charged per pet (in dollars).

(a) Find the revenue earned for each price per pet givenbelow.

$4

$6

$8

(b) Find the price that will yield a maximum revenue.What is the maximum revenue? Explain your results.

85. Graphical Analysis From 1960 to 2003, the per capitaconsumption of cigarettes by Americans (age 18 andolder) can be modeled by

where is the year, with corresponding to 1960.(Source: Tobacco Outlook Report)

(a) Use a graphing utility to graph the model.

(b) Use the graph of the model to approximate themaximum average annual consumption. Beginning in1966, all cigarette packages were required by law tocarry a health warning. Do you think the warning hadany effect? Explain.

(c) In 2000, the U.S. population (age 18 and over) was209,128,094. Of those, about 48,308,590 were smokers.What was the average annual cigarette consumption persmoker in 2000? What was the average daily cigaretteconsumption per smoker?

87. Wind Drag The number of horsepower required toovercome wind drag on an automobile is approximated by

where is the speed of the car (in miles per hour).

(a) Use a graphing utility to graph the function.

(b) Graphically estimate the maximum speed of the car if the power required to overcome wind drag is not to exceed 10 horsepower. Verify your estimate algebraically.

s

0 ≤ s ≤ 100y 0.002s2 0.005s 0.029,

yt 0t

0 ≤ t ≤ 43C 4299 1.8t 1.36t2,

C

p

Rp 12p2 150p

R

p

Rp 25p2 1200p

R

P 230 20x 0.5x2.

xP


86. Data Analysis The numbers (in thousands) of hairdressers and cosmetologists in the United States forthe years 1994 through 2002 are shown in the table.(Source: U.S. Bureau of Labor Statistics)

(a) Use a graphing utility to create a scatter plot of thedata. Let represent the year, with corre-sponding to 1994.

(b) Use the regression feature of a graphing utility tofind a quadratic model for the data.

(c) Use a graphing utility to graph the model in thesame viewing window as the scatter plot. How welldoes the model fit the data?

(d) Use the trace feature of the graphing utility toapproximate the year in which the number of hair-dressers and cosmetologists was the least.

(e) Verify your answer to part (d) algebraically.

(f ) Use the model to predict the number of hairdressersand cosmetologists in 2008.

x 4x

y

Model It

Year Number of hairdressers andcosmetologists, y

1994 753

1995 750

1996 737

1997 748

1998 763

1999 784

2000 820

2001 854

2002 908

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88. Maximum Fuel Economy A study was done to comparethe speed (in miles per hour) with the mileage (in milesper gallon) of an automobile. The results are shown in thetable. (Source: Federal Highway Administration)

(a) Use a graphing utility to create a scatter plot of thedata.

(b) Use the regression feature of a graphing utility to finda quadratic model for the data.

(c) Use a graphing utility to graph the model in the sameviewing window as the scatter plot.

(d) Estimate the speed for which the miles per gallon isgreatest.

Synthesis

True or False? In Exercises 89 and 90, determine whetherthe statement is true or false. Justify your answer.

89. The function given by has no -intercepts.

90. The graphs of

and

have the same axis of symmetry.

91. Write the quadratic function

in standard form to verify that the vertex occurs at

92. Profit The profit (in millions of dollars) for a recre-ational vehicle retailer is modeled by a quadratic functionof the form

where represents the year. If you were president of thecompany, which of the models below would you prefer?Explain your reasoning.

(a) is positive and

(b) is positive and

(c) is negative and

(d) is negative and

93. Is it possible for a quadratic equation to have only one -intercept? Explain.

94. Assume that the function given by

has two real zeros. Show that the -coordinate of thevertex of the graph is the average of the zeros of (Hint:Use the Quadratic Formula.)

Skills Review

In Exercises 95–98, find the equation of the line in slope-intercept form that has the given characteristics.

95. Passes through the points and

96. Passes through the point and has a slope of

97. Passes through the point and is perpendicular to theline

98. Passes through the point and is parallel to the line

In Exercises 99–104, let and let Find the indicated value.

99.

100.

101.

102.

103.

104.

105. Make a Decision To work an extended applicationanalyzing the height of a basketball after it has beendropped, visit this text’s website at college.hmco.com.

g f 0 f g1

fg 1.5

fg47

g f 2 f g3

gx 8x2.fx 14x 3

y 3x 28, 4

4x 5y 100, 3

327

2, 22, 14, 3

f.x

a 0f x ax2 bx c,

x

t ≤ b2a.a

b2a ≤ t.a

t ≤ b2a.a

b2a ≤ t.a

t

P at2 bt c

P

b

2a, f

b2a.

f x ax2 bx c

gx 12x2 30x 1

f x 4x2 10x 7

xf x 12x2 1

yx


Speed, x Mileage, y

15 22.3

20 25.5

25 27.5

30 29.0

35 28.8

40 30.0

45 29.9

50 30.2

55 30.4

60 28.8

65 27.4

70 25.3

75 23.3

332521_0201.qxd 2/8/07 2:56 PM Page 138

Section 2.2 Polynomial Functions of Higher Degree 139

Graphs of Polynomial FunctionsIn this section, you will study basic features of the graphs of polynomial func-tions. The first feature is that the graph of a polynomial function is continuous.Essentially, this means that the graph of a polynomial function has no breaks,holes, or gaps, as shown in Figure 2.10(a). The graph shown in Figure 2.10(b) isan example of a piecewise-defined function that is not continuous.

(a) Polynomial functions have (b) Functions with graphs thatcontinuous graphs. are not continuous are not

polynomial functions.

FIGURE 2.10

The second feature is that the graph of a polynomial function has only smooth,rounded turns, as shown in Figure 2.11. A polynomial function cannot have a sharp turn. For instance, the function given by which has a sharp turn at the point as shown in Figure 2.12, is not a polynomial function.

Polynomial functions have graphs Graphs of polynomial functionswith smooth rounded turns. cannot have sharp turns.FIGURE 2.11 FIGURE 2.12

The graphs of polynomial functions of degree greater than 2 are moredifficult to analyze than the graphs of polynomials of degree 0, 1, or 2. However,using the features presented in this section, coupled with your knowledge ofpoint plotting, intercepts, and symmetry, you should be able to make reasonablyaccurate sketches by hand.

x−4 −3 −2 −1 1 2 3 4

−2

2

3

4

5

6f(x) = x

(0, 0)

y

x

y

0, 0,f x x,

x

y

x

y

What you should learn• Use transformations to

sketch graphs of polynomialfunctions.

• Use the Leading CoefficientTest to determine the endbehavior of graphs of polyno-mial functions.

• Find and use zeros of polyno-mial functions as sketchingaids.

• Use the Intermediate ValueTheorem to help locate zerosof polynomial functions.

Why you should learn itYou can use polynomialfunctions to analyze business situations such as how revenue isrelated to advertising expenses,as discussed in Exercise 98 onpage 151.

Polynomial Functions of Higher Degree

Bill Aron /PhotoEdit, Inc.

2.2

332521_0202.qxd 2/8/07 2:56 PM 9

The polynomial functions that have the simplest graphs are monomials ofthe form where is an integer greater than zero. From Figure 2.13,you can see that when is even, the graph is similar to the graph of and when is odd, the graph is similar to the graph of Moreover, thegreater the value of the flatter the graph near the origin. Polynomial functionsof the form are often referred to as power functions.

(a) If n is even, the graph of (b) If n is odd, the graph of touches the axis at the -intercept. crosses the axis at the -intercept.

FIGURE 2.13

Sketching Transformations of Monomial Functions

Sketch the graph of each function.

a. b.

Solutiona. Because the degree of is odd, its graph is similar to the graph of

In Figure 2.14, note that the negative coefficient has the effect ofreflecting the graph in the -axis.

b. The graph of as shown in Figure 2.15, is a left shift by oneunit of the graph of


Now try Exercise 9.

x

h(x) = (x + 1)

(−2, 1) (0, 1)

(−1, 0)

−2 −1 1

4

3

2

1

y

x

(1, −1)

f(x) = −x5

(−1, 1)

−1 1

−1

1

y

y x4.hx x 14,

xy x3.

f x x5

hx x 14f x x5

xxy xny xn

(1, 1)

(−1, −1)

x−1 1

−1

1y = x 5y = x 3

y

x

(−1, 1)(1, 1)

−1 1

1

2

y = x

y = x

2

4

y

f x xnn,

f x x3.nf x x2,n

nf x xn,


For power functions given byif is even, then

the graph of the function issymmetric with respect to the -axis, and if is odd, then

the graph of the function issymmetric with respect to theorigin.

ny

nf x xn,

Example 1

332521_0202.qxd 2/8/07 2:56 PM 0

The Leading Coefficient TestIn Example 1, note that both graphs eventually rise or fall without bound as moves to the right. Whether the graph of a polynomial function eventually risesor falls can be determined by the function’s degree (even or odd) and by its lead-ing coefficient, as indicated in the Leading Coefficient Test.

x


For each function, identify thedegree of the function andwhether the degree of the func-tion is even or odd. Identify the leading coefficient andwhether the leading coefficientis positive or negative. Use agraphing utility to graph eachfunction. Describe the relation-ship between the degree and thesign of the leading coefficient ofthe function and the right-handand left-hand behavior of thegraph of the function.

a.

b.

c.

d.

e.

f.

g. f x x2 3x 2

f x x4 3x2 2x 1

f x 2x2 3x 4

f x x3 5x 2

f x 2x5 x2 5x 3

f x 2x5 2x2 5x 1

f x x3 2x2 x 1

Exploration

Leading Coefficient TestAs moves without bound to the left or to the right, the graph of thepolynomial function eventually rises orfalls in the following manner.

1. When is odd:

2. When is even:

The dashed portions of the graphs indicate that the test determines only theright-hand and left-hand behavior of the graph.

n

n

f x anxn . . . a1x a0

x

x

f(x) → ∞as x → ∞

f(x) → −∞as x → −∞

y

If the leading coefficient ispositive the graph fallsto the left and rises to the right.

an > 0,

x

f(x) → ∞as x → −∞

f(x) → −∞as x → ∞

y

If the leading coefficient isnegative the graph risesto the left and falls to the right.

an < 0,

x

f(x) → ∞as x → −∞ f(x) → ∞

as x → ∞

y

If the leading coefficient ispositive the graphrises to the left and right.

an > 0,

x

f(x) → −∞as x → −∞

f(x) → −∞as x → ∞

y

If the leading coefficient isnegative the graphfalls to the left and right.

an < 0,

The notation “ as” indicates that the

graph falls to the left. Thenotation “ as ”indicates that the graph rises tothe right.

x →f x →

x → f x →

332521_0202.qxd 2/8/07 2:56 PM Page 141

Applying the Leading Coefficient Test

Describe the right-hand and left-hand behavior of the graph of each function.

a. b. c.

Solution

a. Because the degree is odd and the leading coefficient is negative, the graphrises to the left and falls to the right, as shown in Figure 2.16.

b. Because the degree is even and the leading coefficient is positive, the graphrises to the left and right, as shown in Figure 2.17.

c. Because the degree is odd and the leading coefficient is positive, the graphfalls to the left and rises to the right, as shown in Figure 2.18.


In Example 2, note that the Leading Coefficient Test tells you only whetherthe graph eventually rises or falls to the right or left. Other characteristics of the graph, such as intercepts and minimum and maximum points, must bedetermined by other tests.

Zeros of Polynomial FunctionsIt can be shown that for a polynomial function of degree the followingstatements are true.

1. The function has, at most, real zeros. (You will study this result in detailin the discussion of the Fundamental Theorem of Algebra in Section 2.5.)

2. The graph of has, at most, turning points. (Turning points, also calledrelative minima or relative maxima, are points at which the graph changesfrom increasing to decreasing or vice versa.)

Finding the zeros of polynomial functions is one of the most importantproblems in algebra. There is a strong interplay between graphical and algebraicapproaches to this problem. Sometimes you can use information about the graphof a function to help find its zeros, and in other cases you can use informationabout the zeros of a function to help sketch its graph. Finding zeros of polyno-mial functions is closely related to factoring and finding -intercepts.x

n 1f

nf

n,f

f x x5 xf x x4 5x2 4f x x3 4x


−3 −1 1 3x

1

2

3

yf(x) = −x3 + 4x

FIGURE 2.16

x

y

4−4

4

6

f(x) = x4 − 5x2 + 4

FIGURE 2.17

x−2 2

−2

−1

1

2

yf(x) = x5 − x

FIGURE 2.18

Remember that the zeros of afunction of are the -valuesfor which the function is zero.

xx

A polynomial function is written in standard form if itsterms are written in descendingorder of exponents from left to right. Before applying theLeading Coefficient Test to apolynomial function, it is a good idea to check that thepolynomial function is writtenin standard form.

Example 2

For each of the graphs inExample 2, count the number of zeros of the polynomial func-tion and the number of relativeminima and relative maxima.Compare these numbers withthe degree of the polynomial.What do you observe?

Exploration

332521_0202.qxd 2/8/07 2:56 PM Page 142


Real Zeros of Polynomial FunctionsIf is a polynomial function and is a real number, the following state-ments are equivalent.

1. is a zero of the function

2. is a solution of the polynomial equation

3. is a factor of the polynomial

4. is an -intercept of the graph of f.xa, 0

f x.x a

f x 0.x a

f.x a

af

Finding the Zeros of a Polynomial Function

Find all real zeros of

Then determine the number of turning points of the graph of the function.

f (x) 2x4 2x2.

Example 3

Algebraic SolutionTo find the real zeros of the function, set equal tozero and solve for

Set equal to 0.

Factor completely.

So, the real zeros are and Because the function is a fourth-degree polynomial, thegraph of can have at most turning points.


4 1 3f

x 1.x 1,x 0,

2x2x 1x 1 0

2x2x2 1 0

f x 2x4 2x2 0

x.f x

Graphical SolutionUse a graphing utility to graph In Figure2.19, the graph appears to have zeros at and

Use the zero or root feature, or the zoom and tracefeatures, of the graphing utility to verify these zeros. So, thereal zeros are and From the figure,you can see that the graph has three turning points. This isconsistent with the fact that a fourth-degree polynomial canhave at most three turning points.

FIGURE 2.19

−2

3−3

2

y = −2x4 + 2x2

x 1.x 1,x 0,

1, 0.1, 0,0, 0,

y 2x4 2x2.

In Example 3, note that because is even, the factor yields the repeatedzero The graph touches the -axis at as shown in Figure 2.19.x 0,xx 0.

2x2k

Remove commonmonomial factor.

Repeated ZerosA factor yields a repeated zero of multiplicity

1. If is odd, the graph crosses the -axis at

2. If is even, the graph touches the -axis (but does not cross the -axis)at x a.

xxk

x a.xk

k.x ak > 1,x ak,

332521_0202.qxd 2/8/07 2:56 PM Page 143

To graph polynomial functions, you can use the fact that a polynomialfunction can change signs only at its zeros. Between two consecutive zeros, apolynomial must be entirely positive or entirely negative. This means that whenthe real zeros of a polynomial function are put in order, they divide the realnumber line into intervals in which the function has no sign changes. Theseresulting intervals are test intervals in which a representative -value in theinterval is chosen to determine if the value of the polynomial function is positive(the graph lies above the -axis) or negative (the graph lies below the -axis).

Sketching the Graph of a Polynomial Function

Sketch the graph of

Solution1. Apply the Leading Coefficient Test. Because the leading coefficient is

positive and the degree is even, you know that the graph eventually rises to theleft and to the right (see Figure 2.20).

2. Find the Zeros of the Polynomial. By factoring asyou can see that the zeros of are and (both

of odd multiplicity). So, the -intercepts occur at and Add thesepoints to your graph, as shown in Figure 2.20.

3. Plot a Few Additional Points. Use the zeros of the polynomial to find thetest intervals. In each test interval, choose a representative -value andevaluate the polynomial function, as shown in the table.

4. Draw the Graph. Draw a continuous curve through the points, as shown inFigure 2.21. Because both zeros are of odd multiplicity, you know that thegraph should cross the -axis at and



x

y

−1−2−3−4 2 3 4−1

3

4

5

6

7

f(x) = 3x4 − 4x3

x(0, 0)

Up to rightUp to left

, 043 ))

y

−4 −3 −2 −1 1 2 3 4

7

6

5

4

2

3

−1

x 43.x 0x

x

43, 0.0, 0x

x 43x 0ff x x33x 4,

f x 3x4 4x3

f x 3x4 4x3.

xx

x


Example 4 uses an algebraicapproach to describe the graph of the function. A graphing utilityis a complement to this approach.Remember that an importantaspect of using a graphing utilityis to find a viewing window thatshows all significant features of the graph. For instance, theviewing window in part (a) illus-trates all of the significant featuresof the function in Example 4.

a.

b.

Techno logy

−4 5

−3

3

−2 2

−0.5

0.5

Example 4

If you are unsure of the shapeof a portion of the graph of apolynomial function, plot someadditional points, such as thepoint as shownin Figure 2.21.

0.5, 0.3125

Test interval Representative Value of f Sign Point onx-value graph

Positive

1 Negative

1.5 Positive 1.5, 1.6875f 1.5 1.687543,

1, 1f 1 10, 43

1, 7f 1 71, 0

332521_0202.qxd 2/8/07 2:56 PM Page 144

Sketching the Graph of a Polynomial Function

Sketch the graph of

Solution

1. Apply the Leading Coefficient Test. Because the leading coefficient is nega-tive and the degree is odd, you know that the graph eventually rises to the leftand falls to the right (see Figure 2.22).

2. Find the Zeros of the Polynomial. By factoring

you can see that the zeros of are (odd multiplicity) and (evenmultiplicity). So, the -intercepts occur at and Add these points to your graph, as shown in Figure 2.22.

3. Plot a Few Additional Points. Use the zeros of the polynomial to find the test intervals. In each test interval, choose a representative -value andevaluate the polynomial function, as shown in the table.

4. Draw the Graph. Draw a continuous curve through the points, as shown inFigure 2.23. As indicated by the multiplicities of the zeros, the graph crossesthe -axis at but does not cross the -axis at



1

x−4 −2 −1−3 3 4

f x x x x( ) = 2 + 6− −3 2 92

−2

−1

y

Down to rightUp to left

−2

2

4

6

3

5

x(0, 0)

−4 −2 −1−1

−3 2 31 4

, 032( )

y

32, 0.x0, 0x

x

32, 0.0, 0x

x 32x 0f

12x2x 32

12x4x2 12x 9

f x 2x3 6x2 92 x

f x 2x3 6x2 92x.


Example 5

Observe in Example 5 that thesign of is positive to theleft of and negative to the rightof the zero Similarly, thesign of is negative to theleft and to the right of the zero

This suggests that if thezero of a polynomial function isof odd multiplicity, then the signof changes from one sideof the zero to the other side. Ifthe zero is of even multiplicity,then the sign of does notchange from one side of thezero to the other side.

f x

f x

x 32.

f xx 0.

f x

Test interval Representative Value of f Sign Point onx-value graph

Positive

0.5 Negative

2 Negative 2, 1f 2 132,

0.5, 1f 0.5 10, 32

0.5, 4f 0.5 40.5, 0

332521_0202.qxd 2/8/07 2:56 PM Page 145

The Intermediate Value TheoremThe next theorem, called the Intermediate Value Theorem, illustrates theexistence of real zeros of polynomial functions. This theorem implies that if

and are two points on the graph of a polynomial function suchthat then for any number between and there must be anumber between and such that (See Figure 2.24.)

FIGURE 2.24

The Intermediate Value Theorem helps you locate the real zeros of apolynomial function in the following way. If you can find a value at whicha polynomial function is positive, and another value at which it is nega-tive, you can conclude that the function has at least one real zero between thesetwo values. For example, the function given by is negativewhen and positive when Therefore, it follows from theIntermediate Value Theorem that must have a real zero somewhere between

and as shown in Figure 2.25.

FIGURE 2.25

By continuing this line of reasoning, you can approximate any real zeros ofa polynomial function to any desired accuracy. This concept is further demon-strated in Example 6.

x

f has a zerobetween

2 and 1.− −

f x x x( ) = + + 13 2

( 2, 3)− −

( 1, 1)−

f( 2) = 3− −

f( 1) = 1−

−2 21

−2

−3

−1

y

1,2f

x 1.x 2f x x3 x2 1

x bx a

x

f c d( ) =

f b( )

f a( )

a c b

y

f c d.bacf bf adf a f b,

b, f ba, f a


Intermediate Value TheoremLet and be real numbers such that If is a polynomial functionsuch that then, in the interval takes on every valuebetween and f b.f a

fa, b,f a f b,fa < b.ba

332521_0202.qxd 2/8/07 2:56 PM Page 146

You can use the table feature of a graphing utility to approximatethe zeros of a polynomial function. For instance, for the functiongiven by

create a table that shows the function values for , asshown in the first table at the right. Scroll through the table lookingfor consecutive function values that differ in sign. From the table,you can see that and differ in sign. So, you can concludefrom the Intermediate Value Theorem that the function has a zerobetween 0 and 1. You can adjust your table to show function valuesfor using increments of 0.1, as shown in the second tableat the right. By scrolling through the table you can see that and differ in sign. So, the function has a zero between 0.8 and 0.9. If you repeat this process several times, you should obtain

as the zero of the function. Use the zero or root feature ofa graphing utility to confirm this result.x 0.806

f0.9f0.8

0 ≤ x ≤ 1

f1f0

20 ≤ x ≤ 20

fx 2x3 3x2 3

Techno logy

Approximating a Zero of a Polynomial Function

Use the Intermediate Value Theorem to approximate the real zero of

SolutionBegin by computing a few function values, as follows.

Because is negative and is positive, you can apply the IntermediateValue Theorem to conclude that the function has a zero between and 0. Topinpoint this zero more closely, divide the interval into tenths andevaluate the function at each point. When you do this, you will find that

and

So, must have a zero between and as shown in Figure 2.26. For amore accurate approximation, compute function values between and

and apply the Intermediate Value Theorem again. By continuing thisprocess, you can approximate this zero to any desired accuracy.


f 0.7f 0.8

0.7,0.8f

f 0.7 0.167.f 0.8 0.152

1, 01

f 0f 1

f x x3 x2 1.


x

f has a zero

and 0.7.between 0.8−

−

−1 1 2

−1

2

f x x x( ) = + 13 2−

( 1, 1)− −

(0, 1)(1, 1)

y

FIGURE 2.26

x

0 1

1 1

11

112

f x

Example 6

332521_0202.qxd 2/8/07 2:56 PM Page 147


Exercises 2.2

In Exercises 1– 8, match the polynomial function with itsgraph. [The graphs are labeled (a), (b), (c), (d), (e), (f ), (g), and(h).]

(a) (b)

(c) (d)

(e) (f)

(g) (h)

1. 2.

3. 4.

5. 6.

7. 8.

In Exercises 9–12, sketch the graph of and eachtransformation.

9.

(a) (b)

(c) (d)

10.

(a) (b)

(c) (d)

11.

(a) (b)

(c) (d)

(e) (f) fx 12 x4

2fx 2x4 1

f x 12x 14f x 4 x4

f x x4 3f x x 34

y x4

f x 12x 15f x 1

12x5

f x x5 1f x x 15

y x5

f x x 23 2f x 12x3

f x x3 2f x x 23

y x3

y x n

f x 15x5 2x3

95xf x x4 2x3

f x 13x3 x2

43f x

14x4 3x2

f x 2x3 3x 1f x 2x2 5x

f x x2 4xf x 2x 3

x2

−4

4

−4−2

y

x62

−4

−2−2

y

x42−4

−4

−2

4

y

x84−8

−8

−4−4

8

y

x42−4

−2

6

4

2

y

x84−8

−8

−4−4

8

4

y

x84−8 −4

8

y

x8−8

−8

−4

y


1. The graphs of all polynomial functions are ________, which means that the graphs have no breaks, holes, or gaps.

2. The ________ ________ ________ is used to determine the left-hand and right-hand behavior of the graph of a polynomial function.

3. A polynomial function of degree has at most ________ real zeros and at most ________ turning points.

4. If is a zero of a polynomial function then the following three statements are true.

(a) is a ________ of the polynomial equation

(b) ________ is a factor of the polynomial

(c) is an ________ of the graph

5. If a real zero of a polynomial function is of even multiplicity, then the graph of ________ the -axis at and if it is of odd multiplicity then the graph of ________ the -axis at

6. A polynomial function is written in ________ form if its terms are written in descending order of exponents from left to right.

7. The ________ ________ Theorem states that if is a polynomial function such that then in the interval takes on every value between and f b.f afa, b,

f a f b,f

x a.xfx a,xf

f.a, 0f x.

f x 0.x a

f,x a

n

332521_0202.qxd 2/8/07 2:56 PM Page 148


12.

(a) (b)

(c) (d)

(e) (f)

In Exercises 13–22, describe the right-hand and left-handbehavior of the graph of the polynomial function.

13. 14.

15. 16.

17.

18.

19.

20.

21.

22.

Graphical Analysis In Exercises 23–26, use a graphingutility to graph the functions and in the same viewingwindow. Zoom out sufficiently far to show that theright-hand and left-hand behaviors of and appearidentical.

23.

24.

25.

26.

In Exercises 27– 42, (a) find all the real zeros of thepolynomial function, (b) determine the multiplicity of eachzero and the number of turning points of the graph of thefunction, and (c) use a graphing utility to graph thefunction and verify your answers.

27. 28.

29. 30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

Graphical Analysis In Exercises 43–46, (a) use a graphingutility to graph the function, (b) use the graph to approxi-mate any -intercepts of the graph, (c) set and solvethe resulting equation, and (d) compare the results of part(c) with any -intercepts of the graph.

43.

44.

45.

46.

In Exercises 47–56, find a polynomial function that has thegiven zeros. (There are many correct answers.)

47. 48.

49. 50.

51. 52.

53. 54.

55. 56.

In Exercises 57–66, find a polynomial of degree that hasthe given zero(s). (There are many correct answers.)

Zero(s) Degree

57.

58.

59.

60.

61.

62.

63.

64.

65.

66.

In Exercises 67– 80, sketch the graph of the function by (a)applying the Leading Coefficient Test, (b) finding the zerosof the polynomial, (c) plotting sufficient solution points,and (d) drawing a continuous curve through the points.

67. 68.

69.

70.

71. 72.

73.

74.

75. 76.

77. 78.

79.

80. gx 110x 12x 33

gt 14t 22t 22

hx 13x3x 42f x x2x 4

f x 48x2 3x4f x 5x2 x3

f x 4x3 4x2 15x

f x 3x3 15x2 18x

f x 1 x3f x x3 3x2

gx x2 10x 16

f t 14t 2 2t 15

gx x4 4x2f x x3 9x

n 5x 3, 1, 5, 6

n 5x 0, 4

n 4x 4, 1, 3, 6

n 4x 5, 1, 2

n 3x 9

n 3x 0, 3, 3

n 3x 2, 4, 7

n 3x 3, 0, 1

n 2x 8, 4

n 2x 2

n

2, 4 5, 4 51 3, 1 3

2, 1, 0, 1, 24, 3, 3, 0

0, 2, 50, 2, 3

4, 52, 6

0, 30, 10

y 14x3x2 9

y x5 5x3 4x

y 4x 3 4x 2 8x 8

y 4x3 20x2 25x

x

y 0x

fx x3 4x2 25x 100

gx x3 3x2 4x 12

fx 2x4 2x2 40

fx 5x4 15x2 10

fx x5 x3 6x

gt t 5 6t 3 9t

fx x4 x3 20x2

f t t3 4t2 4t

gx 5xx2 2x 1fx 3x3 12x2 3x

f x 12x2

52x

32

f x 13 x2

13 x

23

fx x2 10x 25ht t 2 6t 9

fx 49 x2fx x2 25

gx 3x4fx 3x4 6x2,

gx x4f x x4 4x3 16x,gx

13x3fx

13x3 3x 2,

gx 3x3f x 3x3 9x 1,

gf

gf

fs 78s3 5s2 7s 1

ht 23t2 5t 3

f x 3x4 2x 5

4

f x 6 2x 4x2 5x3

f x 2x5 5x 7.5

f x 2.1x5 4x3 2

hx 1 x6gx 5 72x 3x2

f x 2x2 3x 1f x 13x3 5x

fx 2x6 1fx 14 x6

2

f x 14x6 1f x x6 4

f x x 26 4f x 18x6

y x6

332521_0202.qxd 2/8/07 2:56 PM Page 149


In Exercises 81–84, use a graphing utility to graph thefunction. Use the zero or root feature to approximate thereal zeros of the function. Then determine the multiplicityof each zero.

81.

82.

83.

84.

In Exercises 85– 88, use the Intermediate Value Theoremand the table feature of a graphing utility to find intervalsone unit in length in which the polynomial function isguaranteed to have a zero. Adjust the table to approximatethe zeros of the function. Use the zero or root feature of agraphing utility to verify your results.

85.

86.

87.

88.

89. Numerical and Graphical Analysis An open boxis to be made from a square piece of material, 36 inches ona side, by cutting equal squares with sides of length fromthe corners and turning up the sides (see figure).

(a) Verify that the volume of the box is given by thefunction

(b) Determine the domain of the function.

(c) Use a graphing utility to create a table that shows thebox height and the corresponding volumes Use thetable to estimate the dimensions that will produce amaximum volume.

(d) Use a graphing utility to graph and use the graph toestimate the value of for which is maximum.Compare your result with that of part (c).

90. Maximum Volume An open box with locking tabs is tobe made from a square piece of material 24 inches on aside. This is to be done by cutting equal squares from thecorners and folding along the dashed lines shown in thefigure.

(a) Verify that the volume of the box is given by thefunction

(b) Determine the domain of the function

(c) Sketch a graph of the function and estimate the value offor which is maximum.

91. Construction A roofing contractor is fabricating guttersfrom 12-inch aluminum sheeting. The contractor plans touse an aluminum siding folding press to create the gutterby creasing equal lengths for the sidewalls (see figure).

(a) Let represent the height of the sidewall of the gutter.Write a function that represents the cross-sectionalarea of the gutter.

(b) The length of the aluminum sheeting is 16 feet. Writea function that represents the volume of one run ofgutter in terms of

(c) Determine the domain of the function in part (b).

(d) Use a graphing utility to create a table that shows thesidewall height and the corresponding volumes Use the table to estimate the dimensions that will pro-duce a maximum volume.

(e) Use a graphing utility to graph Use the graph toestimate the value of for which is a maximum.Compare your result with that of part (d).

(f ) Would the value of change if the aluminum sheetingwere of different lengths? Explain.

x

VxxV.

V.x

x.V

Ax

x 12 − 2x x

Vxx

V.

Vx 8x6 x12 x.

x x x xx x

24 in

.

24 in.

VxxV

V.x

Vx x36 2x2.

x x

x

36 2− x

x

hx x4 10x2 3

gx 3x4 4x3 3

f x 0.11x3 2.07x2 9.81x 6.88

f x x3 3x2 3

hx 15x 223x 52

gx 15x 12x 32x 9

f x 14x4 2x2

f x x3 4x

332521_0202.qxd 2/8/07 2:56 PM Page 150


92. Construction An industrial propane tank is formed byadjoining two hemispheres to the ends of a right circularcylinder. The length of the cylindrical portion of the tank is four times the radius of the hemispherical components (see figure).

(a) Write a function that represents the total volume ofthe tank in terms of

(b) Find the domain of the function.

(c) Use a graphing utility to graph the function.

(d) The total volume of the tank is to be 120 cubic feet.Use the graph from part (c) to estimate the radius andlength of the cylindrical portion of the tank.

Data Analysis: Home Prices In Exercise 93–96, use thetable, which shows the median prices (in thousands ofdollars) of new privately owned U.S. homes in the Midwest

and in the South for the years 1997 through 2003.Thedata can be approximated by the following models.

In the models, represents the year, with correspon-ding to 1997. (Source: U.S. Census Bureau; U.S.Department of Housing and Urban Development)

93. Use a graphing utility to plot the data and graph the modelfor in the same viewing window. How closely does themodel represent the data?

94. Use a graphing utility to plot the data and graph the modelfor in the same viewing window. How closely does themodel represent the data?

95. Use the models to predict the median prices of a newprivately owned home in both regions in 2008. Do youranswers seem reasonable? Explain.

96. Use the graphs of the models in Exercises 93 and 94 towrite a short paragraph about the relationship between themedian prices of homes in the two regions.

98. Revenue The total revenue (in millions of dollars) fora company is related to its advertising expense by thefunction

where is the amount spent on advertising (in tens of thou-sands of dollars). Use the graph of this function, shown inthe figure, to estimate the point on the graph at which thefunction is increasing most rapidly. This point is called thepoint of diminishing returns because any expense abovethis amount will yield less return per dollar invested inadvertising.

Advertising expense(in tens of thousands of dollars)

Rev

enue

(in

mill

ions

of

dolla

rs)

x100 200 300 400

100150

50

200250300350

R

x

0 ≤ x ≤ 400R 1

100,000x3 600x 2,

R

y2

y1

t 7t

y2 0.056t3 1.73t2 23.8t 29

y1 0.139t3 4.42t2 51.1t 39

y2y1

r.V

r

4r

Year, t

7 150 130

8 158 136

9 164 146

10 170 148

11 173 155

12 178 163

13 184 168

y2y1

97. Tree Growth The growth of a red oak tree is approx-imated by the function

where is the height of the tree (in feet) and is its age (in years).

(a) Use a graphing utility to graph the function. (Hint:Use a viewing window in which and

(b) Estimate the age of the tree when it is growingmost rapidly. This point is called the point ofdiminishing returns because the increase in sizewill be less with each additional year.

(c) Using calculus, the point of diminishing returns canalso be found by finding the vertex of the parabolagiven by

Find the vertex of this parabola.

(d) Compare your results from parts (b) and (c).

y 0.009t2 0.274t 0.458.

5 ≤ y ≤ 60.)10 ≤ x ≤ 45

2 ≤ t ≤ 34tG

G 0.003t 3 0.137t2 0.458t 0.839

Model It

332521_0202.qxd 2/8/07 2:56 PM Page 151


Synthesis

True or False? In Exercises 99–101, determine whetherthe statement is true or false. Justify your answer.

99. A fifth-degree polynomial can have five turning points inits graph.

100. It is possible for a sixth-degree polynomial to have onlyone solution.

101. The graph of the function given by

rises to the left and falls to the right.

102. Graphical Analysis For each graph, describe a polyno-mial function that could represent the graph. (Indicate thedegree of the function and the sign of its leadingcoefficient.)

(a) (b)

(c) (d)

103. Graphical Reasoning Sketch a graph of the functiongiven by Explain how the graph of eachfunction differs (if it does) from the graph of eachfunction Determine whether is odd, even, or neither.

(a)

(b)

(c)

(d)

(e)

(f )

(g)

(h)

104. Exploration Explore the transformations of the form

(a) Use a graphing utility to graph the functions given by

and

Determine whether the graphs are increasing ordecreasing. Explain.

(b) Will the graph of always be increasing or decreas-ing? If so, is this behavior determined by or Explain.

(c) Use a graphing utility to graph the function given by

Use the graph and the result of part (b) to determinewhether can be written in the form

Explain.

Skills Review

In Exercises 105–108, factor the expression completely.

105. 106.

107. 108.

In Exercises 109 –112, solve the equation by factoring.

109.

110.

111.

112.

In Exercises 113–116, solve the equation by completing thesquare.

113. 114.

115. 116.

In Exercises 117–122, describe the transformation from acommon function that occurs in Then sketch its graph.

117.

118.

119.

120.

121.

122. f x 10 13x 3

f x 2x 9

f x 7 x 6

f x x 1 5

f x 3 x2

f x x 42

f x.

3x2 4x 9 02x2 5x 20 0

x2 8x 2 0x2 2x 21 0

x2 24x 144 0

12x2 11x 5 0

3x2 22x 16 0

2x2 x 28 0

y3 2164x4 7x3 15x2

6x3 61x2 10x5x2 7x 24

ax h5 k.Hx H

Hx x5 3x3 2x 1.

k?a, h,g

y2 3

5x 25 3.

y1 1

3x 25 1

gx ax h5 k.

gx f f xgx f x34gx

12 f x

gx f 12x

gx fxgx f xgx f x 2gx f x 2

gf.g

f x x4.

x

y

x

y

x

y

x

y

fx 2 x x2 x3 x4 x5 x6 x7

332521_0202.qxd 2/8/07 2:56 PM Page 152

Section 2.3 Polynomial and Synthetic Division 153

What you should learn• Use long division to divide

polynomials by other polynomials.

• Use synthetic division to dividepolynomials by binomials ofthe form .

• Use the Remainder Theoremand the Factor Theorem.

Why you should learn itSynthetic division can help you evaluate polynomial func-tions. For instance, in Exercise 73 on page 160, you will use synthetic division to determinethe number of U.S. militarypersonnel in 2008.

x k

Polynomial and Synthetic Division

© Kevin Fleming/Corbis

2.3

x

( ), 0

( ), 02

1

3

2

y

1 3

−1

−2

−3

1

f(x) = 6x3 − 19x2 + 16x − 4

FIGURE 2.27

Long Division of PolynomialsIn this section, you will study two procedures for dividing polynomials. Theseprocedures are especially valuable in factoring and finding the zeros of polyno-mial functions. To begin, suppose you are given the graph of

Notice that a zero of occurs at as shown in Figure 2.27. Because is a zero of you know that is a factor of This means that thereexists a second-degree polynomial such that

To find you can use long division, as illustrated in Example 1.

Long Division of Polynomials

Divide by and use the result to factor the polyno-mial completely.

Solution

Think

Think

Think

Multiply:

Subtract.

Multiply:

Subtract.

Multiply:

Subtract.

From this division, you can conclude that

and by factoring the quadratic you have

Note that this factorization agrees with the graph shown in Figure 2.27 in that thethree -intercepts occur at and

Now try Exercise 5.

x 23.x 2, x

12,x

6x3 19x2 16x 4 x 22x 13x 2.

6x2 7x 2,

6x3 19x2 16x 4 x 26x2 7x 2

0

2x 2. 2x 4

2x 4

7xx 2. 7x2 14x

7x2 16x

6x2x 2. 6x3 12x2

x 2 ) 6x3 19x2 16x 4

6x2 7x 2

2xx

2.

7x2

x 7x.

6x3

x 6x2.

x 2,6x3 19x2 16x 4

qx,

fx x 2 qx.

qxfx.x 2f,

x 2x 2,f

fx 6x3 19x2 16x 4.

Example 1

332521_0203.qxd 2/8/07 2:57 PM Page 153


In Example 1, is a factor of the polynomial and the long division process produces a remainder of zero. Often, long divisionwill produce a nonzero remainder. For instance, if you divide by

you obtain the following.

Quotient

Divisor Dividend

Remainder

In fractional form, you can write this result as follows.

RemainderDividend

Quotient

Divisor Divisor

This implies that

Multiply each side by

which illustrates the following theorem, called the Division Algorithm.

The Division Algorithm can also be written as

In the Division Algorithm, the rational expression is improper becausethe degree of is greater than or equal to the degree of On the other hand, the rational expression is proper because the degree of is lessthan the degree of dx.

rxrxdxdx.f x

f xdx

f xdx

qx rxdx

.

x 1.x2 3x 5 x 1(x 2 3

x2 3x 5

x 1 x 2

3

x 1

3

2x 2

2x 5

x2 x

x 1 ) x2 3x 5

x 2

x 1,x2 3x 5

6x3 19x2 16x 4,x 2

The Division AlgorithmIf and are polynomials such that and the degree of isless than or equal to the degree of there exist unique polynomials and such that

Dividend Quotient

Divisor Remainder

where or the degree of is less than the degree of If theremainder is zero, divides evenly into f x.dxrx

dx.rxrx 0

f x dxqx rx

rxqxf x,

dxdx 0,dxf x

332521_0203.qxd 2/8/07 2:57 PM Page 154

Before you apply the Division Algorithm, follow these steps.

1. Write the dividend and divisor in descending powers of the variable.

2. Insert placeholders with zero coefficients for missing powers of the variable.


Divide by

SolutionBecause there is no -term or -term in the dividend, you need to line up thesubtraction by using zero coefficients (or leaving spaces) for the missing terms.

So, divides evenly into and you can write


You can check the result of Example 2 by multiplying.


Divide by

Solution

Note that the first subtraction eliminated two terms from the dividend. When thishappens, the quotient skips a term. You can write the result as


2x4 4x3 5x2 3x 2

x2 2x 3 2x2 1

x 1

x2 2x 3.

x 1

x2 2x 3

x2 3x 2

2x4 4x3 6x2

x2 2x 3 ) 2x4 4x3 5x2 3x 2

2x2 1

x2 2x 3.2x4 4x3 5x2 3x 2

x 1x2 x 1 x3 x2 x x2 x 1 x3 1

x 1.x3 1

x 1 x2 x 1,

x3 1,x 1

0

x 1

x 1

x2 x

x2 0x

x3 x2

x 1 ) x3 0x2 0x 1

x2 x 1

xx2

x 1.x3 1


Example 2

Example 3

332521_0203.qxd 2/8/07 2:57 PM Page 155


Synthetic DivisionThere is a nice shortcut for long division of polynomials when dividing bydivisors of the form This shortcut is called synthetic division. The patternfor synthetic division of a cubic polynomial is summarized as follows. (Thepattern for higher-degree polynomials is similar.)

Synthetic division works only for divisors of the form [Rememberthat ] You cannot use synthetic division to divide a polynomialby a quadratic such as

Using Synthetic Division

Use synthetic division to divide by

SolutionYou should set up the array as follows. Note that a zero is included for the missing

-term in the dividend.

1 0 4

Then, use the synthetic division pattern by adding terms in columns and multi-plying the results by

Divisor: Dividend:

Quotient:

So, you have


x4 10x2 2x 4

x 3 x3 3x2 x 1

1

x 3.

x3 3x2 x 1

3 1

1

03

3

109

1

23

1

43

1

x 4 10x2 2x 4x 3

3.

2103

x3

x 3.x4 10x2 2x 4

x2 3.x k x k.

x k.

x k.

k a b c d

ka

a r Remainder

Coefficients of quotient

Coefficients of dividend

Remainder: 1

Synthetic Division (for a Cubic Polynomial)To divide by use the following pattern.

Vertical pattern: Add terms.Diagonal pattern: Multiply by k.

x k,ax3 bx2 cx d

Example 4

332521_0203.qxd 2/8/07 2:57 PM Page 156


The Remainder and Factor TheoremsThe remainder obtained in the synthetic division process has an importantinterpretation, as described in the Remainder Theorem.

For a proof of the Remainder Theorem, see Proofs in Mathematics on page 213.The Remainder Theorem tells you that synthetic division can be used to

evaluate a polynomial function. That is, to evaluate a polynomial function when divide by The remainder will be as illustrated inExample 5.

Using the Remainder Theorem

Use the Remainder Theorem to evaluate the following function at

SolutionUsing synthetic division, you obtain the following.

Because the remainder is you can conclude that

This means that is a point on the graph of You can check this by substituting in the original function.

Check


Another important theorem is the Factor Theorem, stated below. This theo-rem states that you can test to see whether a polynomial has as a factorby evaluating the polynomial at If the result is 0, is a factor.

For a proof of the Factor Theorem, see Proofs in Mathematics on page 213.

x kx k.x k

38 84 10 7 9

f 2 323 822 52 7

x 2f.2, 9

r f kf 2 9.

r 9,

2 3

3

86

2

54

1

72

9

f x 3x3 8x2 5x 7

x 2.

f k,x k.f xx k,f x

The Remainder TheoremIf a polynomial is divided by the remainder is

r f k.

x k,f x

The Factor TheoremA polynomial has a factor if and only if f k 0.x kf x

Example 5

332521_0203.qxd 2/8/07 2:57 PM Page 157


x−4 −1 1 3 4

10

20

30

40

−20

−30

−40

y

(2, 0)

(−1, 0)(−3, 0)

32( (− , 0

f(x) = 2x4 + 7x3 − 4x2 − 27x − 18

FIGURE 2.28

Factoring a Polynomial: Repeated Division

Show that and are factors of

Then find the remaining factors of

SolutionUsing synthetic division with the factor you obtain the following.

Take the result of this division and perform synthetic division again using thefactor

Because the resulting quadratic expression factors as

the complete factorization of is

Note that this factorization implies that has four real zeros:

and

This is confirmed by the graph of which is shown in Figure 2.28.


Throughout this text, the importance of developing several problem-solvingstrategies is emphasized. In the exercises for this section, try using more than onestrategy to solve several of the exercises. For instance, if you find that divides evenly into (with no remainder), try sketching the graph of Youshould find that is an -intercept of the graph.xk, 0

f.f xx k

f,

x 1.x 32,x 3,x 2,

f

fx x 2x 32x 3x 1.

fx

2x2 5x 3 2x 3x 1

3 2

2

116

5

1815

3

99

0

x 3.

2 2

2

74

11

422

18

2736

9

1818

0

x 2,

f x.

fx 2x4 7x3 4x2 27x 18.

x 3x 2

Uses of the Remainder in Synthetic DivisionThe remainder obtained in the synthetic division of by provides the following information.

1. The remainder gives the value of at That is,

2. If is a factor of

3. If is an -intercept of the graph of f.xr 0, k, 0

f x.r 0, x k

r f k.x k.fr

x k,f xr,

0 remainder, so and is a factor.x 3

f 3 0

0 remainder, so andis a factor.x 2

f 2 0

Example 6

332521_0203.qxd 2/8/07 2:57 PM Page 158


Exercises 2.3

Analytical Analysis In Exercises 1 and 2, use long divisionto verify that

1.

2.

Graphical Analysis In Exercises 3 and 4, (a) use a graphingutility to graph the two equations in the same viewingwindow, (b) use the graphs to verify that the expressions are equivalent, and (c) use long division to verify the resultsalgebraically.

3.

4.

In Exercises 5 –18, use long division to divide.

5.

6.

7.

8.

9.

10.

11. 12.

13.

14.

15.

16.

17. 18.

In Exercises 19 –36, use synthetic division to divide.

19.

20.

21.

22.

23.

24.

25.

26.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

In Exercises 37– 44, write the function in the formfor the given value of and demon-

strate that

Function Value of k

37.

38. k 2fx x3 5x2 11x 8

k 4fx x3 x2 14x 11

f k r.k,f x x kqx r

3x3 4x2 5

x 32

4x3 16x2 23x 15

x 12

5 3x 2x2 x3

x 1

180x x4

x 6

3x4

x 2

3x4

x 2

x3 729x 9

x3 512

x 8

x5 13x4 120x 80

x 3

10x4 50x3 800

x 6

5x3 6x 8 x 25x3 6x2 8 x 43x3 16x2 72 x 6x3 75x 250 x 109x3 16x 18x2 32 x 24x3 9x 8x2 18 x 25x3 18x2 7x 6 x 33x3 17x2 15x 25 x 5

2x3 4x2 15x 5

x 12

x4

x 13

x5 7 x3 1x4 3x2 1 x2 2x 3x3 9 x2 16x3 10x2 x 8 2x2 1

8x 5 2x 17x 3 x 2x3 4x2 3x 12 x 3x4 5x3 6x2 x 2 x 26x3 16x2 17x 6 3x 24x3 7x2 11x 5 4x 55x2 17x 12 x 42x2 10x 12 x 3

y2 x 3 2x 4

x2 x 1y1

x3 2x2 5

x2 x 1,

y2 x3 4x 4x

x2 1y1

x5 3x3

x2 1,

y2 x2 8 39

x2 5y1

x4 3x2 1

x2 5,

y2 x 2 4

x 2y1

x2

x 2,

y1 y2.

VOCABULARY CHECK:

1. Two forms of the Division Algorithm are shown below. Identify and label each term or function.

In Exercises 2–5, fill in the blanks.

2. The rational expression is called ________ if the degree of the numerator is greater than or equal to that of the denominator, and is called ________ if the degree of the numerator is less than that of the denominator.

3. An alternative method to long division of polynomials is called ________ ________, in which the divisor must be of the form

4. The ________ Theorem states that a polynomial has a factor if and only if

5. The ________ Theorem states that if a polynomial is divided by the remainder is r f k.x k,f xf k 0.x kf x

x k.

pxqx

f xdx qx

rxdxf x dxqx rx

332521_0203.qxd 2/8/07 2:57 PM Page 159

Function Value of k

39.

40.

41.

42.

43.

44.

In Exercises 45–48, use synthetic division to find each func-tion value. Verify your answers using another method.

45.

(a) (b) (c) (d)

46.

(a) (b) (c) (d)

47.

(a) (b) (c) (d)

48.

(a) (b) (c) (d)

In Exercises 49–56, use synthetic division to show that isa solution of the third-degree polynomial equation, anduse the result to factor the polynomial completely. List allreal solutions of the equation.

Polynomial Equation Value of x

49.

50.

51.

52.

53.

54.

55.

56.

In Exercises 57– 64, (a) verify the given factors of the func-tion (b) find the remaining factors of (c) use your resultsto write the complete factorization of (d) list all real zerosof and (e) confirm your results by using a graphing utilityto graph the function.

Function Factors

57.

58.

59.

60.

Function Factors

61.

62.

63.

64.

Graphical Analysis In Exercises 65–68, (a) use the zero orroot feature of a graphing utility to approximate the zerosof the function accurate to three decimal places, (b) deter-mine one of the exact zeros, and (c) use synthetic divisionto verify your result from part (b), and then factor thepolynomial completely.

65.

66.

67.

68.

In Exercises 69–72, simplify the rational expression byusing long division or synthetic division.

69. 70.

71. 72.x4 9x3 5x2 36x 4

x2 4

x4 6x3 11x2 6x

x2 3x 2

x3 x2 64x 64

x 8

4x3 8x2 x 3

2x 3

f s s3 12s2 40s 24

ht t3 2t2 7t 2

gx x3 4x2 2x 8

f x x3 2x2 5x 10

x 3x 43,f x x3 3x2 48x 144

x52x 1,f x 2x3 x2 10x 5

5x 32x 5,f x 10x3 11x2 72x45

3x 22x 1,f x 6x3 41x2 9x 14

10x 24

x 4x 2,f x 8x4 14x3 71x2

58x 40

x 4x 5,f x x4 4x3 15x2

x 2x 3,f x 3x3 2x2 19x 6

x 1x 2,f x 2x3 x2 5x 2

f,f,f,f,

x 2 5x3 x2 13x 3 0

x 1 3x3 3x2 2 0

x 2x3 2x2 2x 4 0

x 3x3 2x2 3x 6 0

x 2348x3 80x2 41x 6 0

x 122x3 15x2 27x 10 0

x 4x3 28x 48 0

x 2x3 7x 6 0

x

f 10f 5f 2f 1f x 0.4x4 1.6x3 0.7x2 2

h5h2h 13h3

hx 3x3 5x2 10x 1

g1g3g4g2gx x6 4x4 3x2 2

f 8f 12f 2f 1

f x 4x3 13x 10

k 2 2fx 3x3 8x2 10x 8

k 1 3fx 4x3 6x2 12x 4

k 5fx x3 2x2 5x 4

k 2fx x3 3x2 2x 14

k 15fx 10x3 22x2 3x 4

k 23fx 15x4 10x3 6x2 14


73. Data Analysis: Military Personnel The numbers (in thousands) of United States military personnel onactive duty for the years 1993 through 2003 are shownin the table, where represents the year, with corresponding to 1993. (Source: U.S. Department ofDefense)

t 3t

M

Model It

Year, t Military personnel, M

3 1705

4 1611

5 1518

6 1472

7 1439

8 1407

9 1386

10 1384

11 1385

12 1412

13 1434

332521_0203.qxd 2/8/07 2:57 PM Page 160


74. Data Analysis: Cable Television The average monthlybasic rates (in dollars) for cable television in the UnitedStates for the years 1992 through 2002 are shown in thetable, where represents the year, with correspondingto 1992. (Source: Kagan Research LLC)

(a) Use a graphing utility to create a scatter plot of the data.

(b) Use the regression feature of the graphing utility tofind a cubic model for the data. Then graph the modelin the same viewing window as the scatter plot.Compare the model with the data.

(c) Use synthetic division to evaluate the model for theyear 2008.

Synthesis

True or False? In Exercises 75–77, determine whether thestatement is true or false. Justify your answer.

75. If is a factor of some polynomial function thenis a zero of

76. is a factor of the polynomial

77. The rational expression

is improper.

78. Exploration Use the form tocreate a cubic function that (a) passes through the point

and rises to the right, and (b) passes through the pointand falls to the right. (There are many correct

answers.)

Think About It In Exercises 79 and 80, perform thedivision by assuming that n is a positive integer.

79. 80.

81. Writing Briefly explain what it means for a divisor todivide evenly into a dividend.

82. Writing Briefly explain how to check polynomial divi-sion, and justify your reasoning. Give an example.

Exploration In Exercises 83 and 84, find the constant such that the denominator will divide evenly into thenumerator.

83. 84.

Think About It In Exercises 85 and 86, answer thequestions about the division where

85. What is the remainder when Explain.

86. If it is necessary to find is it easier to evaluate thefunction directly or to use synthetic division? Explain.

Skills Review

In Exercises 87–92, use any method to solve the quadraticequation.

87. 88.

89. 90.

91. 92.

In Exercises 93– 96, find a polynomial function that has thegiven zeros. (There are many correct answers.)

93. 94.

95. 96. 2 32 3,1, 2,1 21 2,3,

6, 10, 3, 4

x2 3x 3 02x2 6x 3 0

8x2 22x 15 05x2 3x 14 0

16x2 21 09x2 25 0

f2,k 3?

f x x 32x 3x 13.f x x k,

x5 2x2 x c

x 2

x3 4x2 3x c

x 5

c

x3n 3x2n 5xn 6

xn 2

x3n 9x2n 27xn 27

xn 3

3, 12, 5

f x x kqx r

x3 2x2 13x 10x2 4x 12

6x6 x5 92x4 45x3 184x2 4x 48.

2x 1

f.47

f,7x 4

t 2t

R

Year, t Basic rate, R

2 19.08

3 19.39

4 21.62

5 23.07

6 24.41

7 26.48

8 27.81

9 28.92

10 30.37

11 32.87

12 34.71

Model It (cont inued)

(a) Use a graphing utility to create a scatter plot of thedata.

(b) Use the regression feature of the graphing utility tofind a cubic model for the data. Graph the model inthe same viewing window as the scatter plot.

(c) Use the model to create a table of estimated valuesof Compare the model with the original data.

(d) Use synthetic division to evaluate the model for theyear 2008. Even though the model is relativelyaccurate for estimating the given data, would youuse this model to predict the number of militarypersonnel in the future? Explain.

M.

332521_0203.qxd 2/8/07 2:57 PM Page 161


What you should learn• Use the imaginary unit i to

write complex numbers.

• Add, subtract, and multiplycomplex numbers.

• Use complex conjugates to write the quotient of two complex numbers in standardform.

• Find complex solutions ofquadratic equations.

Why you should learn itYou can use complex numbers to model and solve real-lifeproblems in electronics. Forinstance, in Exercise 83 on page168, you will learn how to usecomplex numbers to find theimpedance of an electricalcircuit.

Complex Numbers

© Richard Megna/Fundamental Photographs

2.4

The Imaginary Unit iYou have learned that some quadratic equations have no real solutions. Forinstance, the quadratic equation has no real solution because there isno real number that can be squared to produce To overcome this deficien-cy, mathematicians created an expanded system of numbers using the imaginaryunit defined as

Imaginary unit

where By adding real numbers to real multiples of this imaginary unit,the set of complex numbers is obtained. Each complex number can be written inthe standard form For instance, the standard form of the complex num-ber is because

In the standard form the real number is called the real part of thecomplex number and the number (where is a real number) is calledthe imaginary part of the complex number.

The set of real numbers is a subset of the set of complex numbers, as shown inFigure 2.29. This is true because every real number can be written as a complexnumber using That is, for every real number you can write

FIGURE 2.29

Realnumbers

Imaginarynumbers

Complexnumbers

a a 0i.a,b 0.a

bbia bi,aa bi,

5 9 5 321 5 31 5 3i.

5 3i5 9a bi.

i2 1.

i 1

i,

1.xx2 1 0

Definition of a Complex NumberIf and are real numbers, the number is a complex number, and itis said to be written in standard form. If the number isa real number. If the number is called an imaginary number.A number of the form where is called a pure imaginary number.b 0,bi,

a bib 0,a bi ab 0,

a biba

Equality of Complex NumbersTwo complex numbers and written in standard form, areequal to each other

Equality of two complex numbers

if and only if and b d.a c

a bi c di

c di,a bi

332521_0204.qxd 2/8/07 2:58 PM Page 162

Operations with Complex NumbersTo add (or subtract) two complex numbers, you add (or subtract) the real andimaginary parts of the numbers separately.

The additive identity in the complex number system is zero (the same as inthe real number system). Furthermore, the additive inverse of the complexnumber is

Additive inverse

So, you have

Adding and Subtracting Complex Numbers

a. Remove parentheses.

Group like terms.


b. Remove parentheses.

Group like terms.

Simplify.


c.

d.


Note in Examples 1(b) and 1(d) that the sum of two complex numbers can bea real number.

0

0 0i

3 4 7 2i i i

3 2i 4 i 7 i 3 2i 4 i 7 i

5i

0 5i

2 2 3i 3i 5i

3i 2 3i 2 5i 3i 2 3i 2 5i

3

3 0

1 4 2i 2i

(1 2i) 4 2i 1 2i 4 2i

5 i

(4 1) (7i 6i)

4 7i 1 6i 4 7i 1 6i

a bi a bi 0 0i 0.

(a bi) a bi.

a bi

Section 2.4 Complex Numbers 163

Addition and Subtraction of Complex NumbersIf and are two complex numbers written in standard form,their sum and difference are defined as follows.

Sum:

Difference: a bi c di a c b di

a bi c di a c b di

c dia bi

Example 1

332521_0204.qxd 2/8/07 2:58 PM Page 163

Many of the properties of real numbers are valid for complex numbers aswell. Here are some examples.

Associative Properties of Addition and Multiplication

Commutative Properties of Addition and Multiplication

Distributive Property of Multiplication Over Addition

Notice below how these properties are used when two complex numbers are multiplied.

Distributive Property


Commutative Property

Associative Property

Rather than trying to memorize this multiplication rule, you should simply remem-ber how the Distributive Property is used to multiply two complex numbers.

Multiplying Complex Numbers

a. Distributive Property

Simplify.

b. Distributive Property


Group like terms.


c. Distributive Property


Simplify.


d. Square of a binomial



Simplify.



5 12i

9 12i 4

i2 1 9 6i 6i 41

9 6i 6i 4i2

33 2i 2i3 2i

3 2i2 3 2i3 2i

13

9 4

i2 1 9 6i 6i 41

9 6i 6i 4i2

(3 2i)(3 2i) 33 2i 2i3 2i

11 2i

8 3 6i 4i

i2 1 8 6i 4i 31

8 6i 4i 3i2

2 i4 3i 24 3i i4 3i

8 12i

42 3i 42 43i

ac bd ad bci

ac bd ad i bci

i2 1 ac ad i bci bd 1

ac adi bci bdi2

a bic di ac di bic di


The procedure described above is similar to multiplying twopolynomials and combining liketerms, as in the FOIL Methodshown in Appendix A.3. Forinstance, you can use the FOILMethod to multiply the two complex numbers from Example 2(b).

F O I L

2 i4 3i 8 6i 4i 3i2

Complete the following.

What pattern do you see? Writea brief description of how youwould find raised to anypositive integer power.

i

i12 i6

i11 i5

i10 i4 1

i9 i3 i

i8 i2 1

i7 i1 i

Exploration

Example 2

332521_0204.qxd 2/8/07 2:58 PM Page 164

Complex ConjugatesNotice in Example 2(c) that the product of two complex numbers can be a realnumber. This occurs with pairs of complex numbers of the form and

called complex conjugates.

Multiplying Conjugates

Multiply each complex number by its complex conjugate.

a. b.

Solutiona. The complex conjugate of is

b. The complex conjugate of is


To write the quotient of and in standard form, where and are not both zero, multiply the numerator and denominator by the complexconjugate of the denominator to obtain

Standard form

Writing a Quotient of Complex Numbersin Standard Form

Expand.

Simplify.



1

10

4

5i

2 16i

20

i2 1 8 6 16i

16 4

8 4i 12i 6i2

16 4i2

2 3i

4 2i

2 3i

4 2i4 2i

4 2i

ac bd bc adi

c2 d2.

a bi

c di

a bi

c dic di

c di

dcc dia bi

25 16 91 16 9i2 4 3i4 3i 42 3i2

4 3i.4 3i

2 1 1 1 i1 i 12 i2

1 i.1 i

4 3i1 i

a2 b2

a2 b21

a bia bi a2 abi abi b2i2

a bi,a bi


Multiply numerator and denominator bycomplex conjugate of denominator.

Note that when you multiply thenumerator and denominator of aquotient of complex numbers by

you are actually multiplying thequotient by a form of 1. You arenot changing the original expres-sion, you are only creating anexpression that is equivalent tothe original expression.

c dic di

Example 3

Example 4

332521_0204.qxd 2/8/07 2:58 PM Page 165

Complex Solutions of Quadratic EquationsWhen using the Quadratic Formula to solve a quadratic equation, you oftenobtain a result such as which you know is not a real number. By factoringout you can write this number in standard form.

The number is called the principal square root of

Writing Complex Numbers in Standard Form

a.

b.

c.


Complex Solutions of a Quadratic Equation

Solve (a) and (b)

Solutiona. Write original equation.

Subtract 4 from each side.

Extract square roots.

b. Write original equation.

Quadratic Formula

Simplify.




1

3±14

3i

56 2 ± 214 i

6

2 ± 56

6

x 2 ± 22 435

23

3x2 2x 5 0

x ±2i

x2 4

x2 4 0

3x2 2x 5 0.x2 4 0

2 23 i

1 23 i 31

12 23 i 3 2i2

1 3 2 1 3 i2

48 27 48 i 27 i 43 i 33 i 3 i

312 3 i12 i 36 i2 61 6

3.3 i

3 31 31 3 i

i 1,3,


Principal Square Root of a Negative NumberIf is a positive number, the principal square root of the negative number

is defined as

a ai.

aaThe definition of principal

square root uses the rule

for and This rule is not valid if both and are negative. For example,

whereas

To avoid problems with squareroots of negative numbers, besure to convert complex num-bers to standard form beforemultiplying.

55 25 5.

5i2 5

25i2

5i5 i

55 5151

bab < 0.a > 0

ab ab

Example 5

Example 6

332521_0204.qxd 2/8/07 2:58 PM Page 166


Exercises 2.4

In Exercises 1– 4, find real numbers and such that theequation is true.

1. 2.

3.

4.

In Exercises 5–16, write the complex number in standardform.

5. 6.

7. 8.

9. 10.

11. 8 12. 45

13. 14.

15. 16.

In Exercises 17–26, perform the addition or subtraction andwrite the result in standard form.

17. 18.

19. 20.

21.

22.

23. 24.

25.

26.

In Exercises 27–36, perform the operation and write theresult in standard form.

27. 28.

29. 30.

31.

32.

33. 34.

35. 36.

In Exercises 37– 44, write the complex conjugate of thecomplex number.Then multiply the number by its complexconjugate.

37. 38.

39. 40.

41. 42.

43. 44.

In Exercises 45–54, write the quotient in standard form.

45. 46.

47. 48.

49. 50.

51. 52.

53. 54.


55. 56.

57. 58.1 i

i

3

4 i

i

3 2i

2i

3 8i

2i

2 i

5

2 i

2

1 i

3

1 i

5i

2 3i2

3i

4 5i 2

8 16i

2i

6 5i

i

6 7i

1 2i

3 i

3 i

5

1 i

2

4 5i

14

2i

5

i

1 88

1520

3 2 i1 5 i

7 12i6 3i

1 2i2 1 2i22 3i2 2 3i2

2 3i2 4 5i2

3 15 i3 15 i

14 10 i14 10 i8i9 4i 6i5 2i 6 2i2 3i 1 i3 2i

1.6 3.2i 5.8 4.3i 3

2 52i 5

3 113 i

22 5 8i 10i13i 14 7i 8 18 4 32 i2 8 5 50

3 2i 6 13i8 i 4 i13 2i 5 6i5 i 6 2i

0.00040.09

4i 2 2i6i i 2

475

1 82 27

3 164 9

a 6 2bi 6 5i

a 1 b 3i 5 8i

a bi 13 4ia bi 10 6i

ba

VOCABULARY CHECK:

1. Match the type of complex number with its definition.

(a) Real Number (i)

(b) Imaginary number (ii)

(c) Pure imaginary number (iii)

In Exercises 2–4, fill in the blanks.

2. The imaginary unit is defined as ________, where ________.

3. If is a positive number, the ________ ________ root of the negative number is defined as

4. The numbers and are called ________ ________, and their product is a real number a2 b2.a bia bi

a a i.aa

i2 i i

b 0a bi,

b 0a 0,a bi,

b 0a 0,a bi,

332521_0204.qxd 2/8/07 2:58 PM Page 167

In Exercises 59–64, write the complex number in standardform.

59. 60.

61. 62.

63. 64.

In Exercises 65–74, use the Quadratic Formula to solve thequadratic equation.

65. 66.

67. 68.

69. 70.

71. 72.

73. 74.

In Exercises 75–82, simplify the complex number and writeit in standard form.

75. 76.

77. 78.

79. 80.

81. 82.

84. Cube each complex number.

(a) 2 (b) (c)

85. Raise each complex number to the fourth power.

(a) 2 (b) (c) (d)

86. Write each of the powers of as or

(a) (b) (c) (d)

Synthesis

True or False? In Exercises 87– 89, determine whether thestatement is true or false. Justify your answer.

87. There is no complex number that is equal to its complexconjugate.

88. is a solution of

89.

90. Error Analysis Describe the error.

91. Proof Prove that the complex conjugate of the productof two complex numbers and is theproduct of their complex conjugates.

92. Proof Prove that the complex conjugate of the sum oftwo complex numbers and is the sum oftheir complex conjugates.

Skills Review


93.

94.

95. 96.

In Exercises 97–100, solve the equation and check yoursolution.

97. 98.

99.

100.

101. Volume of an Oblate Spheroid

Solve for :

102. Newton’s Law of Universal Gravitation

Solve for :

103. Mixture Problem A five-liter container contains amixture with a concentration of 50%. How much of this mixture must be withdrawn and replaced by 100% con-centrate to bring the mixture up to 60% concentration?

F m1m2

r 2r

V 43a2ba

5x 3x 11 20x 15

45x 6 36x 1 0

8 3x 34x 12 19

2x 523x 12x 4

x3 3x2 6 2x 4x 24 3x 8 6x x 2

a2 b2ia1 b1i

a2 b2ia1 b1i

66 66 36 6

i 44 i 150 i 74 i 109 i61 1

x 4 x2 14 56.i6

i 67i 50i 25i 40

1.i, i, 1,i

2i2i2

1 3 i1 3 i

1

2i 3

1

i 3

2 675 3i 35i 5

4i 2 2i 36i 3 i 2

4.5x2 3x 12 01.4x2 2x 10 0

78 x2

34x

516 03

2 x2 6x 9 0

16t 2 4t 3 04x 2 16x 15 0

9x 2 6x 37 04x 2 16x 17 0

x 2 6x 10 0x 2 2x 2 0

2 623 57 10752102

5 106 2


Resistor Inductor Capacitor

Symbol aΩ bΩ cΩ

Impedance cibia

83. Impedance The opposition to current in an electricalcircuit is called its impedance. The impedance in aparallel circuit with two pathways satisfies the equation

where is the impedance (in ohms) of pathway 1 andis the impedance of pathway 2.

(a) The impedance of each pathway in a parallel circuitis found by adding the impedances of all compo-nents in the pathway. Use the table to find and

(b) Find the impedance

20 Ω

10 Ω

16 Ω

9 Ω

1 2

z.

z2.z1

z2

z1

1z

1z1

1z 2

z

Model It

332521_0204.qxd 2/8/07 2:58 PM Page 168

Section 2.5 Zeros of Polynomial Functions 169

The Fundamental Theorem of AlgebraYou know that an th-degree polynomial can have at most real zeros. In thecomplex number system, this statement can be improved. That is, in the complexnumber system, every th-degree polynomial function has precisely zeros. Thisimportant result is derived from the Fundamental Theorem of Algebra, firstproved by the German mathematician Carl Friedrich Gauss (1777–1855).

Using the Fundamental Theorem of Algebra and the equivalence of zerosand factors, you obtain the Linear Factorization Theorem.

For a proof of the Linear Factorization Theorem, see Proofs in Mathematics onpage 214.

Note that the Fundamental Theorem of Algebra and the Linear FactorizationTheorem tell you only that the zeros or factors of a polynomial exist, not how tofind them. Such theorems are called existence theorems.

Zeros of Polynomial Functions

a. The first-degree polynomial has exactly one zero:

b. Counting multiplicity, the second-degree polynomial function

has exactly two zeros: and (This is called a repeated zero.)

c. The third-degree polynomial function

has exactly three zeros: and

d. The fourth-degree polynomial function

has exactly four zeros: and

Now try Exercise 1.

x i.x i,x 1,x 1,

f x x4 1 x 1x 1x ix i

x 2i.x 0, x 2i,

f x x3 4x xx2 4 xx 2ix 2i

x 3.x 3

f x x2 6x 9 x 3x 3

x 2.f x x 2

nn

nn

What you should learn• Use the Fundamental Theorem

of Algebra to determine thenumber of zeros of polynomialfunctions.

• Find rational zeros of polyno-mial functions.

• Find conjugate pairs of com-plex zeros.

• Find zeros of polynomials byfactoring.

• Use Descartes’s Rule of Signsand the Upper and LowerBound Rules to find zeros ofpolynomials.

Why you should learn itFinding zeros of polynomialfunctions is an important part ofsolving real-life problems. Forinstance, in Exercise 112 on page182, the zeros of a polynomialfunction can help you analyzethe attendance at women’scollege basketball games.

Zeros of Polynomial Functions2.5

The Fundamental Theorem of AlgebraIf is a polynomial of degree where then has at least onezero in the complex number system.

fn > 0,n,f x

Linear Factorization TheoremIf is a polynomial of degree where then has precisely linear factors

where are complex numbers.c1, c2, . . . , cn

f x anx c1x c2 . . . x cn

nfn > 0,n,f x

Recall that in order to find thezeros of a function set equal to 0 and solve the resultingequation for For instance, thefunction in Example 1(a) has azero at because

x 2.

x 2 0

x 2

x.

f xf x,

Example 1

332521_0205.qxd 2/8/07 2:58 PM Page 169


−3 −2 21 3x

−3

−2

−1

1

2

3

f(x) = x3 + x + 1y

FIGURE 2.30

The Rational Zero TestThe Rational Zero Test relates the possible rational zeros of a polynomial(having integer coefficients) to the leading coefficient and to the constant term ofthe polynomial.

To use the Rational Zero Test, you should first list all rational numbers whosenumerators are factors of the constant term and whose denominators are factorsof the leading coefficient.

Having formed this list of possible rational zeros, use a trial-and-error method todetermine which, if any, are actual zeros of the polynomial. Note that when theleading coefficient is 1, the possible rational zeros are simply the factors of theconstant term.

Rational Zero Test with Leading Coefficient of 1

Find the rational zeros of

SolutionBecause the leading coefficient is 1, the possible rational zeros are the fac-tors of the constant term. By testing these possible zeros, you can see that neitherworks.

So, you can conclude that the given polynomial has no rational zeros. Note fromthe graph of in Figure 2.30 that does have one real zero between and 0.However, by the Rational Zero Test, you know that this real zero is not a rationalnumber.

Now try Exercise 7.

1ff

1

f 1 13 1 1

3

f 1 13 1 1

±1,

f x x3 x 1.

Possible rational zeros factors of constant term

factors of leading coefficient

The Rational Zero TestIf the polynomial has integer coefficients, every rational zero of has the form

where and have no common factors other than 1, and

q a factor of the leading coefficient an.

p a factor of the constant term a0

qp

Rational zero p

q

fa1x a0f x anxn an1xn1 . . . a2 x 2

Example 2

Historical NoteAlthough they were notcontemporaries, Jean Le Rondd’Alembert (1717–1783)worked independently of Carl Gauss in trying to provethe Fundamental Theorem of Algebra. His efforts weresuch that, in France, theFundamental Theorem ofAlgebra is frequently known as the Theorem of d’Alembert.

Fog

g A

rt M

use

um

332521_0205.qxd 2/8/07 2:58 PM Page 170


Rational Zero Test with Leading Coefficient of 1


SolutionBecause the leading coefficient is 1, the possible rational zeros are the factors ofthe constant term.

Possible rational zeros:

By applying synthetic division successively, you can determine that andare the only two rational zeros.

So, factors as

Because the factor produces no real zeros, you can conclude thatand are the only real zeros of which is verified in Figure 2.31.

FIGURE 2.31


If the leading coefficient of a polynomial is not 1, the list of possible rationalzeros can increase dramatically. In such cases, the search can be shortened inseveral ways: (1) a programmable calculator can be used to speed up the calcula-tions; (2) a graph, drawn either by hand or with a graphing utility, can give a goodestimate of the locations of the zeros; (3) the Intermediate Value Theorem alongwith a table generated by a graphing utility can give approximations of zeros; and(4) synthetic division can be used to test the possible rational zeros.

Finding the first zero is often the most difficult part. After that, the search issimplified by working with the lower-degree polynomial obtained in syntheticdivision, as shown in Example 3.

−4 −2 6x

6

4

−6

8

−8

( 1, 0)− (2, 0)

8−6−8

f x x x x x( ) = + 3 64 3 2− − −

y

f,x 2x 1x2 3

f x x 1x 2x2 3.

f x

2 1

1

2

2

0

3

0

3

6

6

0

1 1

1

1

1

2

1

2

3

3

3

6

6

6

0

x 2x 1

±1, ±2, ±3, ±6

f x x4 x3 x2 3x 6.

Example 3

0 remainder, so is a zero.x 1

0 remainder, so is a zero.x 2

When the list of possible rationalzeros is small, as in Example 2,it may be quicker to test thezeros by evaluating the function.When the list of possible rationalzeros is large, as in Example 3, itmay be quicker to use a differentapproach to test the zeros, suchas using synthetic division orsketching a graph.

332521_0205.qxd 2/8/07 2:58 PM Page 171


Using the Rational Zero Test


SolutionThe leading coefficient is 2 and the constant term is 3.

Possible rational zeros:

By synthetic division, you can determine that is a rational zero.

So, factors as

and you can conclude that the rational zeros of are and


Recall from Section 2.2 that if is a zero of the polynomial function then is a solution of the polynomial equation

Solving a Polynomial Equation

Find all the real solutions of

SolutionThe leading coefficient is and the constant term is

Possible rational solutions:

With so many possibilities (32, in fact), it is worth your time to stop and sketch agraph. From Figure 2.32, it looks like three reasonable solutions would be

and Testing these by synthetic division shows that is the only rational solution. So, you have

Using the Quadratic Formula for the second factor, you find that the two addi-tional solutions are irrational numbers.

and


x 5 265

20 0.5639

x 5 265

20 1.0639

x 210x2 5x 6 0.

x 2x 2.x 12,x

65,

Factors of 12

Factors of 10

±1, ±2, ±3, ±4, ±6, ±12

±1, ±2, ±5, ±10

12.10

10x3 15x2 16x 12 0.

f x 0.x af,x a

x 3.x 12, x 1,f

x 12x 1x 3

f x x 12x2 5x 3

f x

1 2

2

3

2

5

8

5

3

3

3

0

x 1

Factors of 3

Factors of 2

±1, ±3

±1, ±2 ±1, ±3, ±

1

2, ±

3

2

f x 2x3 3x2 8x 3.

x

f x x x x( ) = 10 + 15 + 16 12− −3 2

1

15

5

−10

10

−5

y

FIGURE 2.32

Example 4

Example 5

Remember that when you try to find the rational zeros of apolynomial function with manypossible rational zeros, as inExample 4, you must use trialand error. There is no quickalgebraic method to determinewhich of the possibilities is anactual zero; however, sketchinga graph may be helpful.

332521_0205.qxd 2/8/07 2:58 PM Page 172


Conjugate PairsIn Example 1(c) and (d), note that the pairs of complex zeros are conjugates.That is, they are of the form and

Be sure you see that this result is true only if the polynomial function has realcoefficients. For instance, the result applies to the function given by

but not to the function given by

Finding a Polynomial with Given Zeros

Find a fourth-degree polynomial function with real coefficients that has and as zeros.

SolutionBecause is a zero and the polynomial is stated to have real coefficients, youknow that the conjugate must also be a zero. So, from the Linear FactorizationTheorem, can be written as

For simplicity, let to obtain


Factoring a PolynomialThe Linear Factorization Theorem shows that you can write any th-degreepolynomial as the product of linear factors.

However, this result includes the possibility that some of the values of arecomplex. The following theorem says that even if you do not want to get involvedwith “complex factors,” you can still write as the product of linear and/orquadratic factors. For a proof of this theorem, see Proofs in Mathematics onpage 214.

f x

ci

f x anx c1x c2x c3 . . . x cn

nn

x4 2x3 10x2 18x 9.

f x x2 2x 1x2 9

a 1

f x ax 1x 1x 3ix 3i.

f x3i

3i

3i1,1,

gx x i.x2 1f x

a bi.a bi

Complex Zeros Occur in Conjugate PairsLet be a polynomial function that has real coefficients. If where

is a zero of the function, the conjugate is also a zero of thefunction.

a bib 0,a bi,f x

Factors of a PolynomialEvery polynomial of degree with real coefficients can be written asthe product of linear and quadratic factors with real coefficients, where thequadratic factors have no real zeros.

n > 0

Example 6

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A quadratic factor with no real zeros is said to be prime or irreducible overthe reals. Be sure you see that this is not the same as being irreducible over therationals. For example, the quadratic is irreducible overthe reals (and therefore over the rationals). On the other hand, the quadratic

is irreducible over the rationals but reducibleover the reals.x2 2 x 2 x 2

x2 1 x ix i

In Example 7, if you were not told that is a zero of you could stillfind all zeros of the function by using synthetic division to find the real zeros and 3. Then you could factor the polynomial as Finally, by using the Quadratic Formula, you could determine that the zeros are

and x 1 3i.x 1 3i,x 3,x 2,

x 2x 3x2 2x 10.2

f,1 3i


Find all the zeros of given that is azero of f.

1 3if x x4 3x3 6x2 2x 60

Example 7

Algebraic SolutionBecause complex zeros occur in conjugate pairs, you know that

is also a zero of This means that both

and

are factors of Multiplying these two factors produces

Using long division, you can divide into to obtainthe following.

So, you have

and you can conclude that the zeros of are and


x 2.x 3,x 1 3i,x 1 3i,f

x2 2x 10x 3x 2

f x x2 2x 10x2 x 6

0

6x2 12x 60

6x2 12x 60

x3 2x2 10x

x3 4x2 2x

x4 2x3 10x2

x2 2x 10 ) x4 3x3 6x2 2x 60

x2 x 6

fx2 2x 10

x2 2x 10.

x 12 9i2

x 1 3ix 1 3i x 1 3ix 1 3i

f.

x 1 3ix 1 3i

f.1 3i

Graphical SolutionBecause complex zeros always occur in conju-gate pairs, you know that is also a zero of

Because the polynomial is a fourth-degreepolynomial, you know that there are at most twoother zeros of the function. Use a graphing utilityto graph

as shown in Figure 2.33.

FIGURE 2.33

You can see that and 3 appear to be zeros ofthe graph of the function. Use the zero or rootfeature or the zoom and trace features of thegraphing utility to confirm that and

are zeros of the graph. So, you canconclude that the zeros of are

and x 2.x 3,x 1 3i,x 1 3i,f

x 3x 2

2

−80

−4 5

80

y = x4 − 3x3 + 6x2 + 2x − 60

y x4 3x3 6x2 2x 60

f.1 3i

332521_0205.qxd 2/8/07 2:58 PM Page 174


−4 2 4x

y

5

10

(−2, 0) (1, 0)

f(x) = x5 + x3 + 2x2 −12x + 8

FIGURE 2.34

Example 8 shows how to find all the zeros of a polynomial function,including complex zeros.


Write as the product of linear factors, and listall of its zeros.

SolutionThe possible rational zeros are and Synthetic division producesthe following.

So, you have

You can factor as and by factoring as

you obtain

which gives the following five zeros of

and

From the graph of shown in Figure 2.34, you can see that the real zeros are theonly ones that appear as -intercepts. Note that is a repeated zero.


x 1xf

x 2ix 2i,x 2,x 1,x 1,

f.

f x x 1x 1x 2x 2ix 2i

x 2ix 2i

x2 4 x 4 x 4 x2 4x 1x2 4,x3 x2 4x 4

x 1x 2x3 x2 4x 4.

f x x5 x3 2x2 12x 8

2 1

1

1

2

1

2

2

4

4

8

4

8

8

0

1 1

1

0

1

1

1

1

2

2

2

4

12

4

8

8

8

0

±8.±1, ±2, ±4,

f x x5 x3 2x2 12x 8

1 is a zero.

is a zero.2

In Example 8, the fifth-degreepolynomial function has threereal zeros. In such cases, youcan use the zoom and tracefeatures or the zero or rootfeature of a graphing utility toapproximate the real zeros. Youcan then use these real zeros todetermine the complex zerosalgebraically.

You can use the table feature of a graphing utility to helpyou determine which of the possible rational zeros arezeros of the polynomial in Example 8. The table should beset to ask mode. Then enter each of the possible rationalzeros in the table. When you do this, you will see thatthere are two rational zeros, and 1, as shown at theright.

2

Techno logy

Example 8

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Other Tests for Zeros of PolynomialsYou know that an th-degree polynomial function can have at most real zeros.Of course, many th-degree polynomials do not have that many real zeros. Forinstance, has no real zeros, and has only one realzero. The following theorem, called Descartes’s Rule of Signs, sheds more lighton the number of real zeros of a polynomial.

A variation in sign means that two consecutive coefficients have oppositesigns.

When using Descartes’s Rule of Signs, a zero of multiplicity should becounted as zeros. For instance, the polynomial has two variationsin sign, and so has either two positive or no positive real zeros. Because

you can see that the two positive real zeros are of multiplicity 2.

Using Descartes’s Rule of Signs

Describe the possible real zeros of

SolutionThe original polynomial has three variations in sign.

to to

to

The polynomial

has no variations in sign. So, from Descartes’s Rule of Signs, the polynomialhas either three positive real zeros or one positive

real zero, and has no negative real zeros. From the graph in Figure 2.35, you cansee that the function has only one real zero (it is a positive number, near ).


x 1

f x 3x3 5x2 6x 4

3x3 5x2 6x 4

f x 3x3 5x2 6x 4

f x 3x3 5x2 6x 4

f x 3x3 5x2 6x 4.

x 1

x 1x 1x 2x3 3x 2

x3 3x 2kk

f x x3 1f x x2 1n

nn

Descartes’s Rule of SignsLet be a polynomial withreal coefficients and

1. The number of positive real zeros of is either equal to the number ofvariations in sign of or less than that number by an even integer.

2. The number of negative real zeros of is either equal to the number ofvariations in sign of or less than that number by an even integer.f x

f

f xf

a0 0.f (x) anxn an1x

n1 . . . a2x2 a1x a0

−3 −2 −1 2 3x

y

−3

−2

−1

1

2

3

f(x) = 3x3 − 5x2 + 6x − 4

FIGURE 2.35

Example 9

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Another test for zeros of a polynomial function is related to the sign patternin the last row of the synthetic division array. This test can give you an upper orlower bound of the real zeros of A real number is an upper bound for thereal zeros of if no zeros are greater than Similarly, is a lower bound if noreal zeros of are less than


Find the real zeros of

SolutionThe possible real zeros are as follows.

The original polynomial has three variations in sign. The polynomial

has no variations in sign. As a result of these two findings, you can applyDescartes’s Rule of Signs to conclude that there are three positive real zeros or onepositive real zero, and no negative zeros. Trying produces the following.

So, is not a zero, but because the last row has all positive entries, you knowthat is an upper bound for the real zeros. So, you can restrict the search tozeros between 0 and 1. By trial and error, you can determine that is a zero.So,

Because has no real zeros, it follows that is the only real zero.


x 236x2 3

f x x 2

36x2 3.

x 23

x 1x 1

1 6

6

4

6

2

3

2

5

2

5

3

x 1

6x3 4x2 3x 2

f x 6x3 4x2 3x 2

f x

±1, ±1

2, ±

1

3, ±

1

6, ±

2

3, ±2

Factors of 2

Factors of 6

±1, ±2

±1, ±2, ±3, ±6

f x 6x3 4x2 3x 2.

b.fbb.f

bf.

Upper and Lower Bound RulesLet be a polynomial with real coefficients and a positive leading coeffi-cient. Suppose is divided by using synthetic division.

1. If and each number in the last row is either positive or zero, is anupper bound for the real zeros of

2. If and the numbers in the last row are alternately positive and negative (zero entries count as positive or negative), is a lower boundfor the real zeros of f.

cc < 0

f.cc > 0

x c,fxf x

Example 10

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Before concluding this section, here are two additional hints that can helpyou find the real zeros of a polynomial.

1. If the terms of have a common monomial factor, it should be factored outbefore applying the tests in this section. For instance, by writing

you can see that is a zero of and that the remaining zeros can beobtained by analyzing the cubic factor.

2. If you are able to find all but two zeros of you can always use theQuadratic Formula on the remaining quadratic factor. For instance, if yousucceeded in writing

you can apply the Quadratic Formula to to conclude that the tworemaining zeros are and

Using a Polynomial Model

You are designing candle-making kits. Each kit contains 25 cubic inches ofcandle wax and a mold for making a pyramid-shaped candle. You want the heightof the candle to be 2 inches less than the length of each side of the candle’ssquare base. What should the dimensions of your candle mold be?

SolutionThe volume of a pyramid is where is the area of the base and is theheight. The area of the base is and the height is So, the volume of thepyramid is Substituting 25 for the volume yields the following.

Substitute 25 for

Multiply each side by 3.

Write in general form.

The possible rational solutions are Use synthetic division to test some of the possible solutions. Note that in this case,it makes sense to test only positive -values. Using synthetic division, you candetermine that is a solution.

The other two solutions, which satisfy are imaginary and can be discarded. You can conclude that the base of the candle mold should be 5 inches by 5 inches and the height of the mold should be inches.


5 2 3

x2 3x 15 0,

5 1

1

253

01515

75750

x 5x

±75.±25,±15,±5,±3,x ±1,

0 x3 2x2 75

75 x3 2x2

V. 25 13

x2x 2

V 13 x2x 2.

x 2.x2hBV

13 Bh,

x 2 5.x 2 5x2 4x 1

xx 1x2 4x 1

f x x4 5x3 3x2 x

f x,

fx 0

xx3 5x2 3x 1

f x x4 5x3 3x2 x

f x

Example 11

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Exercises 2.5

In Exercises 1–6, find all the zeros of the function.

1.

2.

3.

4.

5.

6.

In Exercises 7–10, use the Rational Zero Test to listall possible rational zeros of Verify that the zeros of shown on the graph are contained in the list.

7.

8.

9.

10.

In Exercises 11–20, find all the rational zeros of the function.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20. f x 2x4 15x3 23x 2 15x 25

f x 9x4 9x3 58x 2 4x 24

f x 3x3 19x 2 33x 9

Cx 2x3 3x 2 1

px x3 9x 2 27x 27

ht t 3 12t 2 21t 10

hx x3 9x 2 20x 12

gx x3 4x 2 x 4

f x x3 7x 6

f x x3 6x 2 11x 6

x−4 −2 4

2

yf x 4x5 8x4 5x3 10x 2 x 2

x−8 −4 8

−20

−30

−40

yf x 2x4 17x3 35x 2 9x 45

x−6 6 12

18

y

f x x3 4x 2 4x 16

4

2

−4

−4 −2 2x

y

f x x3 3x 2 x 3

ff.

ht t 3t 2t 3i t 3i f x x 6x ix if x x 5x 82

gx) x 2x 43

f x x2x 3x2 1f x xx 62


1. The ________ ________ of ________ states that if is a polynomial of degree then has at least one zero in the complex number system.

2. The ________ ________ ________ states that if is a polynomial of degree then has precisely linear factors where are complex numbers.

3. The test that gives a list of the possible rational zeros of a polynomial function is called the ________ ________ Test.

4. If is a complex zero of a polynomial with real coefficients, then so is its ________,

5. A quadratic factor that cannot be factored further as a product of linear factors containing real numbers is said to be ________ over the ________.

6. The theorem that can be used to determine the possible numbers of positive real zeros and negative real zeros of a function is called ________ ________ of ________.

7. A real number is a(n) ________ bound for the real zeros of if no real zeros are less than and is a(n) ________ bound if no real zeros are greater than b.

b,fb

a bi.a bi

c1, c2, . . . , cnf x anx c1x c2 . . . x cnnfn n > 0,f x

fn n > 0,f x

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In Exercises 21–24, find all real solutions of the polynomialequation.

21.

22.

23.

24.

In Exercises 25–28, (a) list the possible rational zeros of (b) sketch the graph of so that some of the possible zerosin part (a) can be disregarded, and then (c) determine allreal zeros of

25.

26.

27.

28.

In Exercises 29–32, (a) list the possible rational zeros of (b) use a graphing utility to graph so that some of thepossible zeros in part (a) can be disregarded, and then (c) determine all real zeros of

29.

30.

31.

32.

Graphical Analysis In Exercises 33–36, (a) use the zero or root feature of a graphing utility to approximate the zerosof the function accurate to three decimal places, (b) deter-mine one of the exact zeros (use synthetic division to verifyyour result), and (c) factor the polynomial completely.

33. 34.

35.

36.

In Exercises 37– 42, find a polynomial function with realcoefficients that has the given zeros. (There are manycorrect answers.)

37. 38.

39. 40.

41. 42.

In Exercises 43– 46, write the polynomial (a) as the productof factors that are irreducible over the rationals, (b) as theproduct of linear and quadratic factors that are irreducibleover the reals, and (c) in completely factored form.

43.

44.(Hint: One factor is )



In Exercises 47–54, use the given zero to find all the zeros ofthe function.

Function Zero

47.

48.

49.

50.

51.

52.

53.

54.

In Exercises 55–72, find all the zeros of the function andwrite the polynomial as a product of linear factors.

55. 56.

57. 58.

59.

60.

61.

62.

63.

64.

65.

66.

67.

68.

69.

70.

71. 72.

In Exercises 73–78, find all the zeros of the function. Whenthere is an extended list of possible rational zeros, use agraphing utility to graph the function in order to discardany rational zeros that are obviously not zeros of thefunction.

73.

74.

75.

76.

77.

78. gx x 5 8x 4 28x 3 56x 2 64x 32

f x 2x4 5x 3 4x 2 5x 2

f x 9x 3 15x 2 11x 5

f x 16x 3 20x 2 4x 15

f s 2s3 5s2 12s 5

f x x3 24x 2 214x 740

f x x 4 29x 2 100f x x 4 10x 2 9

h x x 4 6x3 10x2 6x 9

gx x 4 4x3 8x2 16x 16

gx 3x3 4x2 8x 8

f x 5x 3 9x 2 28x 6

h x x3 9x2 27x 35

h x x3 x 6

f x x 3 2x 2 11x 52

gx x3 6x2 13x 10

h(x) x 3 3x 2 4x 2

f z z 2 2z 2

f y y4 625

f x x 4 81

gx x 2 10x 23hx x 2 4x 1

f x x 2 x 56f x x 2 25

1 3if x x 3 4x 2 14x 20

3 2 if x x 4 3x 3 5x 2 21x 22

1 3 ih x 3x 3 4x 2 8x 8

3 ig x 4x 3 23x 2 34x 10

5 2ig x x 3 7x 2 x 87

2if x 2x 4 x 3 7x 2 4x 4

3if x x3 x 2 9x 9

5if x 2x 3 3x 2 50x 75

x 2 4.f x x 4 3x 3 x 2 12x 20

x 2 2x 2.f x x 4 4x 3 5x 2 2x 6

x 2 6.f x x 4 2x 3 3x 2 12x 18

f x x 4 6x 2 27

5, 5, 1 3 i23, 1, 3 2 i

2, 4 i, 4 i6, 5 2i, 5 2i

4, 3i, 3i1, 5i, 5i

gx 6x4 11x3 51x 2 99x 27

hx x5 7x4 10x3 14x 2 24x

Pt t 4 7t 2 12f x x 4 3x 2 2

f x 4x3 7x 2 11x 18

f x 32x3 52x 2 17x 3

f x 4x4 17x 2 4

f x 2x4 13x 3 21x 2 2x 8

f.

ff,

f x 4x3 12x 2 x 15

f x 4x3 15x 2 8x 3

f x 3x3 20x 2 36x 16

f x x3 x 2 4x 4

f.

ff,

x5 x4 3x3 5x 2 2x 0

2y4 7y 3 26y 2 23y 6 0

x 4 13x 2 12x 0

z4 z3 2z 4 0

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In Exercises 79– 86, use Descartes’s Rule of Signs to deter-mine the possible numbers of positive and negative zerosof the function.

79. 80.

81. 82.

83.

84.

85.

86.

In Exercises 87– 90, use synthetic division to verify theupper and lower bounds of the real zeros of

87.

(a) Upper: (b) Lower:

88.


89.


90.


In Exercises 91–94, find all the real zeros of the function.

91.

92.

93.

94.

In Exercises 95–98, find all the rational zeros of the polyno-mial function.

95.

96.

97.

98.

In Exercises 99–102, match the cubic function with thenumbers of rational and irrational zeros.

(a) Rational zeros: 0; irrational zeros: 1(b) Rational zeros: 3; irrational zeros: 0(c) Rational zeros: 1; irrational zeros: 2(d) Rational zeros: 1; irrational zeros: 0

99. 100.

101. 102.

103. Geometry An open box is to be made from a rectangularpiece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides.

(a) Let represent the length of the sides of the squaresremoved. Draw a diagram showing the squaresremoved from the original piece of material and theresulting dimensions of the open box.

(b) Use the diagram to write the volume of the box asa function of Determine the domain of the function.

(c) Sketch the graph of the function and approximate thedimensions of the box that will yield a maximumvolume.

(d) Find values of such that Which of thesevalues is a physical impossibility in the constructionof the box? Explain.

104. Geometry A rectangular package to be sent by adelivery service (see figure) can have a maximumcombined length and girth (perimeter of a cross section)of 120 inches.

(a) Show that the volume of the package is

(b) Use a graphing utility to graph the function andapproximate the dimensions of the package that willyield a maximum volume.

(c) Find values of such that Which ofthese values is a physical impossibility in theconstruction of the package? Explain.

105. Advertising Cost A company that produces MP3players estimates that the profit (in dollars) for selling aparticular model is given by

where is the advertising expense (in tens of thousands of dollars). Using this model, find the smaller of twoadvertising amounts that will yield a profit of $2,500,000.

106. Advertising Cost A company that manufactures bicy-cles estimates that the profit (in dollars) for selling aparticular model is given by

where is the advertising expense (in tens of thousands of dollars). Using this model, find the smaller of twoadvertising amounts that will yield a profit of $800,000.

x

0 ≤ x ≤ 50P 45x 3 2500x 2 275,000,

P

x

0 ≤ x ≤ 60P 76x 3 4830x 2 320,000,

P

V 13,500.x

Vx 4x 230 x.

y

x

x

V 56.x

x.V

x

f x x3 2xf x x3 x

f x x3 2f x x3 1

f z z3 116 z2

12z

13

166z311z2 3z 2

f x x3 14 x 2 x

14

144x3 x 2 4x 1

f x x332 x2

232 x 6

122x33x 2 23x 12

Px x 4 254 x 2 9

144x4 25x 2 36

gx 3x3 2x 2 15x 10

f y 4y3 3y 2 8y 6

f z 12z3 4z 2 27z 9

f x 4x3 3x 1

x 4x 3

f x 2x4 8x 3

x 3x 5

f x x4 4x3 16x 16

x 3x 4

f x 2x3 3x 2 12x 8

x 1x 4

f x x4 4x3 15

f.

f x 3x3 2x 2 x 3

f x 5x3 x 2 x 5

f x 4x3 3x 2 2x 1

gx 2x3 3x 2 3

hx 2x4 3x 2hx 3x4 2x 2 1

hx 4x2 8x 3gx 5x5 10x

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107. Geometry A bulk food storage bin with dimensions 2 feet by 3 feet by 4 feet needs to be increased in size tohold five times as much food as the current bin. (Assumeeach dimension is increased by the same amount.)

(a) Write a function that represents the volume of thenew bin.

(b) Find the dimensions of the new bin.

108. Geometry A rancher wants to enlarge an existingrectangular corral such that the total area of the newcorral is 1.5 times that of the original corral. The currentcorral’s dimensions are 250 feet by 160 feet. The rancherwants to increase each dimension by the same amount.

(a) Write a function that represents the area of the newcorral.

(b) Find the dimensions of the new corral.

(c) A rancher wants to add a length to the sides of thecorral that are 160 feet, and twice the length to thesides that are 250 feet, such that the total area of thenew corral is 1.5 times that of the original corral.Repeat parts (a) and (b). Explain your results.

109. Cost The ordering and transportation cost (inthousands of dollars) for the components used in manu-facturing a product is given by

where is the order size (in hundreds). In calculus, it canbe shown that the cost is a minimum when

Use a calculator to approximate the optimal order size tothe nearest hundred units.

110. Height of a Baseball A baseball is thrown upward froma height of 6 feet with an initial velocity of 48 feet persecond, and its height (in feet) is

where is the time (in seconds). You are told the ballreaches a height of 64 feet. Is this possible?

111. Profit The demand equation for a certain product iswhere is the unit price (in dollars)

of the product and is the number of units produced andsold. The cost equation for the product is

where is the total cost (in dollars)and is the number of units produced. The total profitobtained by producing and selling units is

You are working in the marketing department of the com-pany that produces this product, and you are asked todetermine a price that will yield a profit of 9 milliondollars. Is this possible? Explain.

Synthesis

True or False? In Exercises 113 and 114, decide whetherthe statement is true or false. Justify your answer.

113. It is possible for a third-degree polynomial function withinteger coefficients to have no real zeros.

114. If is a zero of the function given by

then must also be a zero of

Think About It In Exercises 115–120, determine (if possi-ble) the zeros of the function if the function has zeros at

and

115. 116. gx 3f xgx f x

x r3.x r1, x r2,fg

f.x i

f x x 3 ix2 ix 1

x i

p

P R C xp C.

xx

CC 80x 150,000,

xpp 140 0.0001x,

t

ht 16t 2 48t 6, 0 ≤ t ≤ 3

h

3x3 40x 2 2400x 36,000 0.

x

x ≥ 1C 100200

x 2

x

x 30,

C

A

V

112. Athletics The attendance (in millions) at NCAAwomen’s college basketball games for the years 1997through 2003 is shown in the table, where representsthe year, with corresponding to 1997.(Source: National Collegiate Athletic Association)

(a) Use the regression feature of a graphing utility tofind a cubic model for the data.

(b) Use the graphing utility to create a scatter plot ofthe data. Then graph the model and the scatterplot in the same viewing window. How do theycompare?

(c) According to the model found in part (a), in whatyear did attendance reach 8.5 million?

(d) According to the model found in part (a), in whatyear did attendance reach 9 million?

(e) According to the right-hand behavior of themodel, will the attendance continue to increase?Explain.

t 7t

A

Model It

Year, t Attendance, A

7 6.7

8 7.4

9 8.0

10 8.7

11 8.8

12 9.5

13 10.2

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117. 118.

119. 120.

121. Exploration Use a graphing utility to graph the func-tion given by for different values of

Find values of such that the zeros of satisfy thespecified characteristics. (Some parts do not have uniqueanswers.)

(a) Four real zeros

(b) Two real zeros, each of multiplicity 2

(c) Two real zeros and two complex zeros

(d) Four complex zeros

122. Think About It Will the answers to Exercise 121change for the function

(a) (b)

123. Think About It A third-degree polynomial function has real zeros and 3, and its leading coefficient isnegative. Write an equation for Sketch the graph of How many different polynomial functions are possible for

124. Think About It Sketch the graph of a fifth-degreepolynomial function whose leading coefficient is positiveand that has one zero at of multiplicity 2.

125. Writing Compile a list of all the various techniques forfactoring a polynomial that have been covered so far inthe text. Give an example illustrating each technique, andwrite a paragraph discussing when the use of eachtechnique is appropriate.

126. Use the information in the table to answer each question.

(a) What are the three real zeros of the polynomial func-tion

(b) What can be said about the behavior of the graph of at

(c) What is the least possible degree of Explain. Canthe degree of ever be odd? Explain.

(d) Is the leading coefficient of positive or negative?Explain.

(e) Write an equation for (There are many correctanswers.)

(f) Sketch a graph of the equation you wrote inpart (e).

127. (a) Find a quadratic function (with integer coefficients)that has as zeros. Assume that is a positiveinteger.

(b) Find a quadratic function (with integer coefficients)that has as zeros. Assume that is a positiveinteger.

128. Graphical Reasoning The graph of one of the following functions is shown below. Identify the functionshown in the graph. Explain why each of the others is notthe correct function. Use a graphing utility to verify yourresult.

(a)

(b)

(c)

(d)

Skills Review

In Exercises 129–132, perform the operation and simplify.

129.

130.

131.

132.

In Exercises 133–138, use the graph of f to sketch the graphof g. To print an enlarged copy of the graph, go to thewebsite www.mathgraphs.com.

133.

134.

135.

136.

137.

138. gx f 12xgx f 2xgx f xgx 2 f xgx f x 2

x(−2, 0)

(4, 4)

(2, 2)

(0, 2)

21 43

4

5

f

ygx f x 2

9 5i9 5i6 2i1 7i12 5i 16i

3 6i 8 3i

2 4

–40

–30

–20

10

x

y

k x x 1)x 2x 3.5h x x 2)x 3.5x 2 1g x x 2)x 3.5f x x 2x 2)x 3.5

ba ± bif

b±bif

f.

f

ff ?

x 1?f

f ?

x 3

f ?

f.f.

12,2,

f

gx f 2xgx f x 2g?

fkk.f x x 4 4x 2 k

gx f xgx 3 f xgx f 2xgx f x 5

Interval Value of

Positive

Negative

Negative

Positive4,

1, 4

2, 1

, 2

f x

332521_0205.qxd 2/8/07 2:58 PM Page 183


What you should learn• Find the domains of rational

functions.

• Find the horizontal and verticalasymptotes of graphs of rational functions.

• Analyze and sketch graphs ofrational functions.

• Sketch graphs of rational functions that have slantasymptotes.

• Use rational functions to modeland solve real-life problems.

Why you should learn itRational functions can be usedto model and solve real-lifeproblems relating to business.For instance, in Exercise 79 onpage 196, a rational function isused to model average speedover a distance.

Rational Functions2.6

IntroductionA rational function can be written in the form

where and are polynomials and is not the zero polynomial.In general, the domain of a rational function of includes all real numbers

except -values that make the denominator zero. Much of the discussion ofrational functions will focus on their graphical behavior near the -values excludedfrom the domain.

Finding the Domain of a Rational Function

Find the domain of and discuss the behavior of near any excluded

-values.

SolutionBecause the denominator is zero when the domain of is all real numbersexcept To determine the behavior of near this excluded value, evaluate

to the left and right of as indicated in the following tables.

From the table, note that as approaches 0 from the left, decreases withoutbound. In contrast, as approaches 0 from the right, increases withoutbound. Because decreases without bound from the left and increases without bound from the right, you can conclude that is not continuous. Thegraph of is shown in Figure 2.36.

FIGURE 2.36

Now try Exercise 1.

x−1 1 2

1

2

−1

f x( ) = 1x

y

ff

fxf xxf xx

x 0,f xfx 0.

fx 0,

x

ff x 1x

xx

xDxDxNx

f x N(x)

D(x)

Example 1

x 0

10001001021f x

0.0010.010.10.51

x 0 0.001 0.01 0.1 0.5 1

1000 100 10 2 1f x

Mike Powell /Getty Images

Use the table and trace featuresof a graphing utility to verifythat the function inExample 1 is not continuous.

fx 1x

Exploration

332521_0206.qxd 2/8/07 2:59 PM Page 184

Horizontal and Vertical AsymptotesIn Example 1, the behavior of near is denoted as follows.

as as

decreases without bound increases without boundas approaches 0 from the left. as approaches 0 from the right.

The line is a vertical asymptote of the graph of as shown in Figure 2.37.From this figure, you can see that the graph of also has a horizontal asymptote—the line This means that the values of approach zero as increases or decreases without bound.

0 as 0 as

approaches 0 as approaches 0 as decreases without bound. increases without bound.

Eventually (as or ), the distance between the horizon-tal asymptote and the points on the graph must approach zero. Figure 2.38 showsthe horizontal and vertical asymptotes of the graphs of three rational functions.

The graphs of in Figure 2.37 and inFigure 2.38(a) are hyperbolas. You will study hyperbolas in Section 10.4.

f x 2x 1x 1f x 1x

xx

xf xxf x

xf xxf x

xf x 1xy 0.f

f,x 0

xxf xf x

0xf x0xf x

x 0f

Section 2.6 Rational Functions 185

x

f(x) = 2x + 1x + 1

Horizontalasymptote:

y = 2

Verticalasymptote:

x = −1

−2−3 1

1

3

4

y

−1

2

(a)

FIGURE 2.38

x−2 −1 1 2

1

3

2

f(x) = 4x + 12


y = 0

y

(b)

x32−1

2

3

4

f (x) =(x −1)2

Verticalasymptote:

x = 1Horizontalasymptote:

y = 0

y2

1

(c)

x

f(x) = 1x

−1−2 1 2

1

2

−1 Horizontalasymptote:

y = 0

Verticalasymptote:

x = 0

y

FIGURE 2.37

Definitions of Vertical and Horizontal Asymptotes

1. The line is a vertical asymptote of the graph of if

or

as either from the right or from the left.

2. The line is a horizontal asymptote of the graph of if

as or .xx

bf x

fy b

a,x

f xf x

fx aUse a table of values to determine whether the functionsin Figure 2.38 are continuous. If the graph of a function has an asymptote, can you concludethat the function is not continuous? Explain.

Exploration

332521_0206.qxd 2/8/07 2:59 PM Page 185

Finding Horizontal and Vertical Asymptotes

Find all horizontal and vertical asymptotes of the graph of each rational function.

a. b.

Solution

a. For this rational function, the degree of the numerator is equal to the degreeof the denominator. The leading coefficient of the numerator is 2 and the lead-ing coefficient of the denominator is 1, so the graph has the line as ahorizontal asymptote. To find any vertical asymptotes, set the denominatorequal to zero and solve the resulting equation for

Set denominator equal to zero.

Factor.

Set 1st factor equal to 0.

Set 2nd factor equal to 0.

This equation has two real solutions and so the graph has thelines and as vertical asymptotes. The graph of the function isshown in Figure 2.39.

b. For this rational function, the degree of the numerator is equal to the degree of the denominator. The leading coefficient of both the numerator and denom-inator is 1, so the graph has the line as a horizontal asymptote. To findany vertical asymptotes, first factor the numerator and denominator as follows.

By setting the denominator (of the simplified function) equal to zero,you can determine that the graph has the line as a vertical asymptote.

Now try Exercise 9.

x 3x 3

f x x2 x 2x2 x 6

x 1x 2x 2x 3

x 1x 3

, x 2

y 1

x 1x 1x 1,x 1

x 1 x 1 0

x 1 x 1 0

x 1x 1 0

x2 1 0

x.

y 2

f x x2 x 2x2 x 6

f x 2x2

x2 1


Asymptotes of a Rational FunctionLet be the rational function given by

where and have no common factors.

1. The graph of has vertical asymptotes at the zeros of

2. The graph of has one or no horizontal asymptote determined bycomparing the degrees of and

a. If the graph of has the line (the -axis) as a horizontalasymptote.

b. If the graph of has the line (ratio of the leadingcoefficients) as a horizontal asymptote.

c. If the graph of has no horizontal asymptote.fn > m,

y anbmfn m,

xy 0fn < m,

Dx.Nxf

Dx.f

DxNx

f x NxDx

anxn an1x

n1 . . . a1x a0

bmxm bm1xm1 . . . b1x b0

f

Verticalasymptote:

x = −1

Horizontalasymptote: y = 2

Verticalasymptote:

x = 1

x

y

4

3

2

1

−2−3−4 2 3 4

2x2

x2 − 1f (x) =

1−1

FIGURE 2.39

Example 2

332521_0206.qxd 2/8/07 2:59 PM Page 186


Analyzing Graphs of Rational FunctionsTo sketch the graph of a rational function, use the following guidelines.

Guidelines for Analyzing Graphs of Rational FunctionsLet where and are polynomials.

1. Simplify if possible.

2. Find and plot the -intercept (if any) by evaluating

3. Find the zeros of the numerator (if any) by solving the equationThen plot the corresponding -intercepts.

4. Find the zeros of the denominator (if any) by solving the equationThen sketch the corresponding vertical asymptotes.

5. Find and sketch the horizontal asymptote (if any) by using the rule forfinding the horizontal asymptote of a rational function.

6. Plot at least one point between and one point beyond each -intercept andvertical asymptote.

7. Use smooth curves to complete the graph between and beyond thevertical asymptotes.

x

Dx 0.

xNx 0.

f 0.y

f,

DxNxf x NxDx,

Some graphing utilities have difficulty graphing rational functions that have vertical asymptotes. Often, the utility will connect parts of the graph that are not supposed to be connected. For instance, the topscreen on the right shows the graph of

Notice that the graph should consist of two unconnected portions—one to the left of and the other to the right of To eliminate this problem, you can try changing the mode of the graphing utility to dot mode. The problem with this is that the graph is then represented as a collection of dots (as shown in the bottom screen on the right) rather thanas a smooth curve.

−5

−5

5

5x 2.x 2

f x 1

x 2.

−5

−5

5

5

Techno logy

You may also want to test for symmetry when graphingrational functions, especially forsimple rational functions. Recallfrom Section 1.6 that the graph of

is symmetric with respect to theorigin.

f x 1

x

The concept of test intervals from Section 2.2 can be extended to graphingof rational functions. To do this, use the fact that a rational function can changesigns only at its zeros and its undefined values (the -values for which itsdenominator is zero). Between two consecutive zeros of the numerator andthe denominator, a rational function must be entirely positive or entirely negative.This means that when the zeros of the numerator and the denominator of arational function are put in order, they divide the real number line into testintervals in which the function has no sign changes. A representative -value ischosen to determine if the value of the rational function is positive (the graph liesabove the -axis) or negative (the graph lies below the -axis).xx

x

x

332521_0206.qxd 2/8/07 2:59 PM Page 187


Sketching the Graph of a Rational Function

Sketch the graph of and state its domain.

Solutiony-intercept: because

x-intercept: None, because

Vertical asymptote: zero of denominator

Horizontal asymptote: because degree of degree of

Additional points:

By plotting the intercepts, asymptotes, and a few additional points, you can obtainthe graph shown in Figure 2.40. The domain of is all real numbers except



Sketch the graph of

and state its domain.

Solutiony-intercept: None, because is not in the domain

x-intercept: because



Additional points:

By plotting the intercepts, asymptotes, and a few additional points, you can obtainthe graph shown in Figure 2.41. The domain of is all real numbers except


x 0.xf

DxNx y 2,

x 0,

2x 1 012, 0,

x 0

f x 2x 1

x

x 2.xg

DxNx <y 0,

x 2,

3 0

g0 320, 3

2,

gx 3

x 2You can use transformations to help you sketch graphs ofrational functions. For instance,the graph of in Example 3 is avertical stretch and a right shiftof the graph of because

3f x 2.

3 1x 2

gx 3

x 2

f x 1x

g

4 6

2

4

−2

−4

x

Verticalasymptote:

x = 2


y = 0

g(x) =x − 2

3y

2

FIGURE 2.40

3

3 4

1

1 2−1

−1

−2

−4 −3 −2x

f x( ) = x2 1x −


= 2y

Verticalasymptote:

= 0x

y

2

FIGURE 2.41

Test Representative Value of g Sign Point oninterval x-value graph

Negative

3 Positive 3, 3g3 32,

4, 0.5g4 0.54, 2

Test Representative Value of f Sign Point oninterval x-value graph

Positive

Negative

4 Positive 4, 1.75f 4 1.7512,

14, 2f 1

4 2140, 12

1, 3f 1 31, 0

Example 3

Example 4

332521_0206.qxd 2/8/07 2:59 PM Page 188


Sketch the graph of

Solution

Factoring the denominator, you have

y-intercept: because

x-intercept:

Vertical asymptotes: zeros of denominator


Additional points:

The graph is shown in Figure 2.42.


A Rational Function with Common Factors

Sketch the graph of

SolutionBy factoring the numerator and denominator, you have


x-intercept: because

Vertical asymptote: zero of (simplified) denominator


Additional points:

The graph is shown in Figure 2.43. Notice that there is a hole in the graph atbecause the function is not defined when


x 3.x 3

DxNx y 1,

x 1,

f 3 03, 0,f 0 30, 3,

f x x2 9

x2 2x 3

x 3x 3x 3x 1

x 3

x 1, x 3.

f x x2 9x2 2x 3.

DxNx <y 0,

x 2,x 1,

0, 0f 0 00, 0,

f x x

x 1x 2.

f x xx2 x 2.


3

−3

−2

−1

−

1

2

3

x

Verticalasymptote:

x = 2

Verticalasymptote:

x = −1


y = 0

f(x) =x2 − x − 2

x

y

1 2

FIGURE 2.42

y

x

x2 − 9x2 − 2x − 3

f(x) =


y = 1

−3−4 1 2 3 4 5 6

2

3

−2

−3

−4

−5

Verticalasymptote:

x = −1

1

−1

FIGURE 2.43 HOLE AT x 3


Negative

Positive

1 Negative

3 Positive 3, 0.75f 3 0.752,

1, 0.5f 1 0.50, 2

0.5, 0.4f 0.5 0.40.51, 0

3, 0.3f 3 0.33, 1


Positive

Negative

2 Positive 2, 1.67f 2 1.671,

2, 1f 2 123, 1

4, 0.33f 4 0.334, 3

Example 5

If you are unsure of the shapeof a portion of the graph of arational function, plot someadditional points. Also note that when the numerator and the denominator of a rationalfunction have a common factor,the graph of the function has ahole at the zero of the commonfactor (see Example 6).

Example 6

332521_0206.qxd 2/8/07 2:59 PM Page 189


Slant AsymptotesConsider a rational function whose denominator is of degree 1 or greater. If thedegree of the numerator is exactly one more than the degree of the denominator,the graph of the function has a slant (or oblique) asymptote. For example, thegraph of

has a slant asymptote, as shown in Figure 2.44. To find the equation of a slantasymptote, use long division. For instance, by dividing into youobtain

Slant asymptote

As increases or decreases without bound, the remainder term approaches 0, so the graph of approaches the line as shown inFigure 2.44.

A Rational Function with a Slant Asymptote

Sketch the graph of

SolutionFactoring the numerator as allows you to recognize the -intercepts. Using long division

allows you to recognize that the line is a slant asymptote of the graph.


x-intercepts: and


Slant asymptote:

Additional points:

The graph is shown in Figure 2.45.


y x

x 1,

2, 01, 0

f 0 20, 2,

y x

f x x2 x 2

x 1 x

2

x 1

xx 2x 1

f x x2 x 2x 1.

y x 2,f2x 1x

y x 2

x 2 2

x 1.f x

x2 x

x 1

x2 x,x 1

f x x2 x

x 1x

y = x − 2

Slantasymptote:

Verticalasymptote:

= 1x −

yf x( ) =

x + 1x x2 −

−2−4−6−8 2 4 6 8−2

−4

FIGURE 2.44

4

4

5

5

3

3

2

−2

−2

−3

−3x

Verticalasymptote:

x = 1

Slantasymptote:

y = x

f(x) =x − 1

x2 − x − 2

y

11

FIGURE 2.45

Example 7


Negative

0.5 Positive

1.5 Negative

3 Positive 3, 2f 3 22,

1.5, 2.5f 1.5 2.51, 2

0.5, 4.5f 0.5 4.51, 1

2, 1.33f 2 1.332, 1

332521_0206.qxd 2/8/07 2:59 PM Page 190

ApplicationsThere are many examples of asymptotic behavior in real life. For instance,Example 8 shows how a vertical asymptote can be used to analyze the cost ofremoving pollutants from smokestack emissions.

Cost-Benefit Model

A utility company burns coal to generate electricity. The cost (in dollars) ofremoving % of the smokestack pollutants is given by

for Sketch the graph of this function. You are a member of a statelegislature considering a law that would require utility companies to remove 90%of the pollutants from their smokestack emissions. The current law requires 85%removal. How much additional cost would the utility company incur as a result ofthe new law?

SolutionThe graph of this function is shown in Figure 2.46. Note that the graph has avertical asymptote at Because the current law requires 85% removal,the current cost to the utility company is

Evaluate when

If the new law increases the percent removal to 90%, the cost will be

Evaluate when

So, the new law would require the utility company to spend an additional

FIGURE 2.46


Cos

t (in

thou

sand

s of

dol

lars

)

20 40 60 80 100

200

400

600

800

1000

Percent of pollutants removed

p

C

C =80,000 p100 − p

90%

85%

Smokestack Emissions

Subtract 85% removal costfrom 90% removal cost.720,000 453,333 $266,667.

p 90.C $720,000. C 80,000(90)

100 90

p 85.C $453,333. C 80,000(85)

100 85

p 100.

0 ≤ p < 100.

C 80,000p100 p

pC


Example 8

332521_0206.qxd 2/8/07 2:59 PM Page 191


If you go on to take a course in calculus, you will learn an analytic techniquefor finding the exact value of that produces a minimum area. In this case, thatvalue is x 62 8.485.

x

Finding a Minimum Area

A rectangular page is designed to contain 48 square inches of print. The marginsat the top and bottom of the page are each 1 inch deep. The margins on each sideare inches wide. What should the dimensions of the page be so that the leastamount of paper is used?

112

Example 9

Graphical SolutionLet be the area to be minimized. From Figure 2.47,you can write

The printed area inside the margins is modeled byor To find the minimum area,

rewrite the equation for in terms of just one variableby substituting for

The graph of this rational function is shown in Figure2.48. Because represents the width of the printedarea, you need consider only the portion of the graphfor which is positive. Using a graphing utility, youcan approximate the minimum value of to occurwhen inches. The corresponding value of is

inches. So, the dimensions should be

by

FIGURE 2.48


00

24

A =(x + 3)(48 + 2x)

x , x > 0

200

y 2 7.6 inches.x 3 11.5 inches

488.5 5.6yx 8.5

Ax

x

x > 0 x 348 2x

x,

A x 348

x 2

y.48xA

y 48x.48 xy

A x 3y 2.

A

Numerical SolutionLet be the area to be minimized. From Figure 2.47, you canwrite

The printed area inside the margins is modeled by orTo find the minimum area, rewrite the equation for

in terms of just one variable by substituting for

Use the table feature of a graphing utility to create a table ofvalues for the function

beginning at From the table, you can see that the mini-mum value of occurs when is somewhere between 8 and9, as shown in Figure 2.49. To approximate the minimum valueof to one decimal place, change the table so that it starts at

and increases by 0.1. The minimum value of occurswhen as shown in Figure 2.50. The correspondingvalue of is inches. So, the dimensions shouldbe inches by inches.


y 2 7.6x 3 11.5488.5 5.6y

x 8.5,y1x 8

y1

xy1

x 1.

y1 x 348 2x

x

x 348 2x

x, x > 0

A x 348x

2y.48xA

y 48x.48 xy

A x 3 y 2.

A

1 in.

1 in.

11 in.211 in.2y

x

FIGURE 2.47

332521_0206.qxd 2/8/07 2:59 PM Page 192


Exercises 2.6

In Exercises 1– 4, (a) complete each table for the function,(b) determine the vertical and horizontal asymptotes of the graph of the function, and (c) find the domain of thefunction.

1. 2.

3. 4.

In Exercises 5 –12, (a) find the domain of the function, (b)decide if the function is continuous, and (c) identify anyhorizontal and vertical asymptotes.

5. 6.

7. 8.

9. 10.

11. 12.

In Exercises 13 –16, match the rational function with itsgraph. [The graphs are labeled (a), (b), (c), and (d).]

(a) (b)

(c) (d)

13. 14.

15. 16.

In Exercises 17–20, find the zeros (if any) of the rationalfunction.

17. 18.

19. 20. gx x3 8

x 2 1f x 1

3

x 3

hx 2 5

x 2 2gx

x 2 1

x 1

f x x 2

x 4f x

x 1

x 4

f x 1

x 5f x

2

x 3

x

4

−4

−2−2

−4

2

y

x

4

2

−264

y

x

4

2

−2

−4

−2−4−6−8

y

x

4

2

−2

−4

6

y

f x 3x 2 x 5

x 2 1f x

3x 2 1

x 2 x 9

f x 2x 2

x 1f x

x 3

x 2 1

f x 1 5x

1 2xf x

2 x

2 x

f x 4

x 23f x

1

x 2

x4

4

−8 8

8

y

x4−4

−4−8

−8

8

y

f x 4x

x2 1f x

3x 2

x 2 1

x−8

−4−4 4 8

8

12

y

x−4

−4

−2 2 4−2

2

4

y

f x 5x

x 1f x

1

x 1


1. Functions of the form where and are polynomials and is not the zero polynomial, are called ________ ________.

2. If as from the left or the right, then is a ________ ________ of the graph of

3. If as then is a ________ ________ of the graph of

4. For the rational function given by if the degree of is exactly one more than the degree of then the graph of has a ________ (or oblique) ________.fDx,

Nxf x NxDx,f.y bx → ±,f x → b

f.x ax → af x → ±

DxDxNxf x NxDx,

x

0.5

0.9

0.99

0.999

f x x

1.5

1.1

1.01

1.001

f x x

5

10

100

1000

f x

332521_0206.qxd 2/8/07 2:59 PM Page 193


In Exercises 21– 26, find the domain of the function andidentify any horizontal and vertical asymptotes.

21. 22.

23. 24.

25. 26.

In Exercises 27–46, (a) find the domain of the function,(b) decide if the function is continuous, (c) identify all intercepts, (d) identify any horizontal and vertical asymptotes, and (e) plot additional solution points asneeded to sketch the graph of the function.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39.

40.

41. 42.

43. 44.

45. 46.

Analytical, Numerical, and Graphical Analysis In Exercises47– 50, do the following.

(a) Determine the domains of and

(b) Simplify and find any vertical asymptotes of thegraph of

(c) Compare the functions by completing the table.

(d) Use a graphing utility to graph and in the sameviewing window.

(e) Explain why the graphing utility may not show thedifference in the domains of and

47.

48.

49.

50.

In Exercises 51–64, (a) state the domain of the function, (b)identify all intercepts, (c) identify any vertical and slantasymptotes, and (d) plot additional solution points asneeded to sketch the graph of the rational function.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60.

61. 62. f x 2x 2 5x 5

x 2f x

x 2 x 1

x 1

gx x 3

2x 2 8f x

x 3

x 2 1

f x x2

3x 1f t

t 2 1

t 5

hx x2

x 1gx

x2 1x

f x 1 x2

xf x

2x2 1x

gx x2 5

xhx

x2 4

x

gx 2

x 4f x

2x 6

x 2 7x 12,

gx 1

xf x

x 2

x 2 2x,

gx xf x x 2x 2x 2 2x

,

gx x 1f x x 2 1

x 1,

g.f

gf

f.f

g.f

f x x2 16x 4

f t t2 1t 1

f x 3x2 8x 42x2 3x 2

f x 2x2 5x 22x2 x 6

f x 5x 4

x2 x 12f x

x2 3xx2 x 6

f x x2 x 2

x 3 2x2 5x 6

f x 2x2 5x 3

x 3 2x2 x 2

gx x2 2x 8

x2 9hx

x2 5x 4x2 4

f x 1

x 22gs

s

s 2 1

f t 1 2t

tf x

x 2

x 2 9

Px 1 3x

1 xCx

5 2x

1 x

gx 1

3 xhx

1

x 2

f x 1

x 3f x

1

x 2

f x 6x2 11x 36x2 7x 3

f x x2 3x 42x2 x 1

f x x2 4

x2 3x 2f x

x2 1x2 2x 3

f x x 3x2 9

f x x 4

x2 16


x 0 1 1.5 2 2.5 3

gx

f x

1

x 0 1 2 3 4 5 6

gx

f x

x 0 0.5 1 1.5 2 3

gx

f x

0.5

x 0 1

gx

f x

0.511.523

332521_0206.qxd 2/8/07 2:59 PM Page 194


63.

64.

In Exercises 65– 68, use a graphing utility to graph therational function. Give the domain of the function andidentify any asymptotes. Then zoom out sufficiently far sothat the graph appears as a line. Identify the line.

65.

66.

67.

68.

Graphical Reasoning In Exercises 69–72, (a) use the graphto determine any -intercepts of the graph of the rationalfunction and (b) set and solve the resulting equationto confirm your result in part (a).

69. 70.

71. 72.

73. Pollution The cost (in millions of dollars) of removingof the industrial and municipal pollutants discharged

into a river is given by

(a) Use a graphing utility to graph the cost function.

(b) Find the costs of removing 10%, 40%, and 75% of thepollutants.

(c) According to this model, would it be possible toremove 100% of the pollutants? Explain.

74. Recycling In a pilot project, a rural township is givenrecycling bins for separating and storing recyclableproducts. The cost (in dollars) for supplying bins to of the population is given by


(b) Find the costs of supplying bins to 15%, 50%, and 90%of the population.

(c) According to this model, would it be possible to supplybins to 100% of the residents? Explain.

75. Population Growth The game commission introduces100 deer into newly acquired state game lands. The popu-lation of the herd is modeled by

where is the time in years (see figure).

(a) Find the populations when and

(b) What is the limiting size of the herd as time increases?

76. Concentration of a Mixture A 1000-liter tank contains50 liters of a 25% brine solution. You add liters of a 75%brine solution to the tank.

(a) Show that the concentration , the proportion of brineto total solution, in the final mixture is

(b) Determine the domain of the function based on thephysical constraints of the problem.

(c) Sketch a graph of the concentration function.

(d) As the tank is filled, what happens to the rate at whichthe concentration of brine is increasing? What percentdoes the concentration of brine appear to approach?

C 3x 50

4x 50.

C

x

t 25.t 10,t 5,

Time (in years)

Dee

r po

pula

tion

t50 100 150 200

200

400

600

800

1000

1200

1400

N

t

t ≥ 0N 205 3t1 0.04t

,

N

0 ≤ p < 100.C 25,000p

100 p ,

p%C

0 ≤ p < 100.C 255p

100 p ,

p%C

x−8 −4

−44 8

4

8

y

x−4

−4

−2 4

2

4

y

y x 3 2

xy

1

x x

x−2 8642

−4

2

4

6

y

x−2 864

−4

2

4

6

y

y 2x

x 3y

x 1

x 3

y 0x

hx 12 2x x 2

24 x

gx 1 3x 2 x 3

x 2

f x 2x 2 x

x 1

f x x 2 5x 8

x 3

f x 2x3 x2 8x 4

x2 3x 2

f x 2x3 x2 2x 1

x2 3x 2

332521_0206.qxd 2/8/07 3:00 PM Page 195

77. Page Design A page that is inches wide and incheshigh contains 30 square inches of print. The top and bottommargins are 1 inch deep and the margins on each side are 2 inches wide (see figure).

(a) Show that the total area on the page is

(b) Determine the domain of the function based on thephysical constraints of the problem.

(c) Use a graphing utility to graph the area function andapproximate the page size for which the least amountof paper will be used. Verify your answer numericallyusing the table feature of the graphing utility.

78. Page Design A rectangular page is designed to contain64 square inches of print. The margins at the top andbottom of the page are each 1 inch deep. The margins oneach side are inches wide. What should the dimensionsof the page be so that the least amount of paper is used?

80. Sales The sales (in millions of dollars) for the YankeeCandle Company in the years 1998 through 2003 are shownin the table. (Source: The Yankee Candle Company)

1998 184.5 1999 256.6 2000 338.8

2001 379.8 2002 444.8 2003 508.6

A model for these data is given by

where represents the year, with corresponding to1998.

(a) Use a graphing utility to plot the data and graph themodel in the same viewing window. How well does themodel fit the data?

(b) Use the model to estimate the sales for the YankeeCandle Company in 2008.

(c) Would this model be useful for estimating sales after2008? Explain.

Synthesis


81. A polynomial can have infinitely many vertical asymptotes.

82. The graph of a rational function can never cross one of itsasymptotes.

Think About It In Exercises 83 and 84, write a rationalfunction that has the specified characteristics. (There aremany correct answers.)

83. Vertical asymptote: None

Horizontal asymptote:

84. Vertical asymptote:

Horizontal asymptote: None

Skills Review

In Exercises 85– 88, completely factor the expression.

85. 86.

87. 88.

In Exercises 89–92, solve the inequality and graph thesolution on the real number line.

89. 90.

91. 92.

93. Make a Decision To work an extended application analyzing the total manpower of the Department of Defense,visit this text’s website at college.hmco.com. (DataSource: U.S. Department of Defense)

122x 3 ≥ 54x 2 < 20

5 2x > 5x 110 3x ≤ 0

x 3 6x2 2x 12x 3 5x2 4x 20

3x2 23x 36x2 15x 56

x 1x 2,

y 2

f

t 8t

S 5.816t2 130.680.004t2 1.00

, 8 ≤ t ≤ 13

S

112

A 2xx 11

x 4.

A

x

y

1 in.

1 in.

2 in. 2 in.

yx

196 Chapter 2 Polynomial and Rational Functions196 Chapter 2 Polynomial and Rational Functions

79. Average Speed A driver averaged 50 miles per houron the round trip between Akron, Ohio, and Columbus,Ohio, 100 miles away. The average speeds for goingand returning were and miles per hour, respectively.

(a) Show that

(b) Determine the vertical and horizontal asymptotesof the graph of the function.

(c) Use a graphing utility to graph the function.

(d) Complete the table.

(e) Are the results in the table what you expected?Explain.

(f) Is it possible to average 20 miles per hour in onedirection and still average 50 miles per hour on theround trip? Explain.

y 25x

x 25.

yx

Model It

x 30 35 40 45 50 55 60

y

332521_0206.qxd 2/8/07 3:00 PM Page 196

Section 2.7 Nonlinear Inequalities 197

What you should learn• Solve polynomial inequalities.

• Solve rational inequalities.

• Use inequalities to model andsolve real-life problems.

Why you should learn itInequalities can be used tomodel and solve real-lifeproblems. For instance, inExercise 73 on page 205, apolynomial inequality is used to model the percent of house-holds that own a television andhave cable in the United States.

Nonlinear Inequalities

© Jose Luis Pelaez, Inc. /Corbis

2.7

Polynomial InequalitiesTo solve a polynomial inequality such as you can use the factthat a polynomial can change signs only at its zeros (the -values that make thepolynomial equal to zero). Between two consecutive zeros, a polynomial must beentirely positive or entirely negative. This means that when the real zeros of apolynomial are put in order, they divide the real number line into intervals inwhich the polynomial has no sign changes. These zeros are the critical numbersof the inequality, and the resulting intervals are the test intervals for the inequal-ity. For instance, the polynomial above factors as

and has two zeros, and These zeros divide the real number lineinto three test intervals:

and (See Figure 2.51.)

So, to solve the inequality you need only test one value fromeach of these test intervals to determine whether the value satisfies the originalinequality. If so, you can conclude that the interval is a solution of the inequality.

FIGURE 2.51 Three test intervals for

You can use the same basic approach to determine the test intervals for anypolynomial.

x2 2x 3

x−2−3−4 −1 0 1 2 3 5

Test Interval( , 1)− −

Test Interval( , 3)−1

Test Interval(3, )

4

Zero= 1x −

Zero= 3x

x2 2x 3 < 0,

3, .1, 3,, 1,

x 3.x 1

x2 2x 3 x 1x 3

xx2 2x 3 < 0,

Finding Test Intervals for a PolynomialTo determine the intervals on which the values of a polynomial are entirelynegative or entirely positive, use the following steps.

1. Find all real zeros of the polynomial, and arrange the zeros in increasingorder (from smallest to largest). These zeros are the critical numbers ofthe polynomial.

2. Use the critical numbers of the polynomial to determine its test intervals.

3. Choose one representative -value in each test interval and evaluate thepolynomial at that value. If the value of the polynomial is negative, thepolynomial will have negative values for every -value in the interval. Ifthe value of the polynomial is positive, the polynomial will have positivevalues for every -value in the interval.x

x

x

332521_0207.qxd 2/8/07 3:00 PM Page 197


Solving a Polynomial Inequality

Solve

SolutionBy factoring the polynomial as

you can see that the critical numbers are and So, thepolynomial’s test intervals are

and Test intervals

In each test interval, choose a representative -value and evaluate the polynomial.

Test Interval x-Value Polynomial Value Conclusion

Positive

Negative

Positive

From this you can conclude that the inequality is satisfied for all -values inThis implies that the solution of the inequality is the

interval as shown in Figure 2.52. Note that the original inequality contains a less than symbol. This means that the solution set does not contain theendpoints of the test interval

FIGURE 2.52


As with linear inequalities, you can check the reasonableness of a solutionby substituting -values into the original inequality. For instance, to check thesolution found in Example 1, try substituting several -values from the interval

into the inequality

Regardless of which -values you choose, the inequality should be satisfied.You can also use a graph to check the result of Example 1. Sketch the graph

of as shown in Figure 2.53. Notice that the graph is below the -axis on the interval 2, 3.x

y x2 x 6,

x

x2 x 6 < 0.

2, 3x

x

x

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7

Choose = 3.( + 2)( 3) > 0

xx x

−−

Choose =( + 2)( 3) > 0

xx x

4.−

Choose =( + 2)( 3) < 0

xx x

0.−

2, 3.

2, 3,x2 x 6 < 02, 3.

x

42 4 6 6x 43,

02 0 6 6x 02, 3

32 3 6 6x 3, 2

x

3, .2, 3,, 2,

x 3.x 2

x2 x 6 x 2x 3

x2 x 6 < 0.

x2 51−1 4

2

1

−6

−7

−2

−3

−3−4

y x x= 62 − −

y

FIGURE 2.53

Example 1

332521_0207.qxd 2/8/07 3:00 PM Page 198


In Example 1, the polynomial inequality was given in general form (with thepolynomial on one side and zero on the other). Whenever this is not the case, youshould begin the solution process by writing the inequality in general form.

Solving a Polynomial Inequality

Solve

SolutionBegin by writing the inequality in general form.

Write original inequality.


Factor.

The critical numbers are and and the test intervals are and

Test Interval x-Value Polynomial Value Conclusion

Negative

Positive

Negative

Positive

From this you can conclude that the inequality is satisfied on the open intervalsand Therefore, the solution set consists of all real numbers in the

intervals and as shown in Figure 2.54.

FIGURE 2.54


When solving a polynomial inequality, be sure you have accounted for theparticular type of inequality symbol given in the inequality. For instance, inExample 2, note that the original inequality contained a “greater than” symboland the solution consisted of two open intervals. If the original inequality hadbeen

the solution would have consisted of the closed interval and the interval4, .

4, 322x3 3x2 32x ≥ 48

4, ,4, 324, .4, 32

253 352 325 48x 54,

223 322 322 48x 232, 4

203 302 320 48x 04, 32253 352 325 48x 5, 4

4, ., 4, 4, 32, 32, 4,

x 4,x 4, x 32,

x 4x 42x 3 > 0

2x3 3x2 32x 48 > 0

2x3 3x2 32x > 48

2x3 3x2 32x > 48.

−6

x

−5 −4 −3 −2 −1 0 1 2 3 4 5 6−7

Choose =( 4)( + 4)(2 3) > 0

xx x x

0.− −

Choose = 5.( 4)( + 4)(2 3) > 0

xx x x− −

Choose =( 4)( + 4)(2 3) < 0

xx x x

−5.− −

Choose = 2( 4)( + 4)(2 3) < 0

xx x x

.− −

You may find it easier to deter-mine the sign of a polynomialfrom its factored form. Forinstance, in Example 2, if thetest value is substitutedinto the factored form

you can see that the sign patternof the factors is

which yields a negative result.Try using the factored forms ofthe polynomials to determinethe signs of the polynomials inthe test intervals of the otherexamples in this section.

x 4x 42x 3

x 2

Example 2

332521_0207.qxd 2/8/07 3:00 PM Page 199


Each of the polynomial inequalities in Examples 1 and 2 has a solution setthat consists of a single interval or the union of two intervals. When solving theexercises for this section, watch for unusual solution sets, as illustrated inExample 3.

Unusual Solution Sets

a. The solution set of the following inequality consists of the entire set of realnumbers, In other words, the value of the quadratic is positive for every real value of

b. The solution set of the following inequality consists of the single real numberbecause the quadratic has only one critical number,

and it is the only value that satisfies the inequality.

c. The solution set of the following inequality is empty. In other words, the quad-ratic is not less than zero for any value of

d. The solution set of the following inequality consists of all real numbers except In interval notation, this solution set can be written as


x2 4x 4 > 0

, 2 2, .x 2.

x2 3x 5 < 0

x.x2 3x 5

x2 2x 1 ≤ 0

x 1,x2 2x 11,

x2 2x 4 > 0

x.x2 2x 4, .

You can use a graphing utility to verify the results in Example 3. Forinstance, the graph of is shown below. Notice that the -values are greater than 0 for all values of as stated in Example 3(a).

Use the graphing utility to graph the following:

Explain how you can use the graphs to verify the results of parts (b), (c),and (d) of Example 3.

−2

−9 9

10

y x2 4x 4y x2 3x 5y x2 2x 1

x,yy x2 2x 4

Exploration

Example 3

332521_0207.qxd 2/8/07 3:00 PM Page 200


Rational InequalitiesThe concepts of critical numbers and test intervals can be extended to rationalinequalities. To do this, use the fact that the value of a rational expression canchange sign only at its zeros (the -values for which its numerator is zero) and itsundefined values (the -values for which its denominator is zero). These twotypes of numbers make up the critical numbers of a rational inequality. Whensolving a rational inequality, begin by writing the inequality in general form withthe rational expression on the left and zero on the right.

Solving a Rational Inequality

Solve

Solution

Write original inequality.


Find the LCD and add fractions.

Simplify.

Critical numbers: Zeros and undefined values of rational expression

Test intervals:

Test: Is

After testing these intervals, as shown in Figure 2.55, you can see that theinequality is satisfied on the open intervals and Moreover,because when you can conclude that the solutionset consists of all real numbers in the intervals (Be sure to usea closed interval to indicate that can equal 8.)

FIGURE 2.55


x

4 5 6 7 8 9

Choose = 6.x

> 0

Choose = 9.xChoose = 4.x

< 0< 0−x + 8x − 5

−x + 8x − 5

−x + 8x − 5

x, 5 8, .

x 8,x 8x 5 08, .(, 5)

x 8

x 5≤ 0?

, 5, 5, 8, 8,

x 5, x 8

x 8

x 5≤ 0

2x 7 3x 15

x 5≤ 0

2x 7

x 5 3 ≤ 0

2x 7

x 5≤ 3

2x 7

x 5≤ 3.

xx

Example 4

332521_0207.qxd 2/8/07 3:00 PM Page 201


Number of units sold(in millions)

Prof

it (i

n m

illio

ns o

f do

llars

)

0 2 4 6 8 10

x

P

−100

−50

0

50

100

150

200

Calculators

FIGURE 2.57

Number of units sold(in millions)

Rev

enue

(in

mill

ions

of

dolla

rs)

0 2 4 6 8x

R

10

50

100

150

200

250

Calculators

FIGURE 2.56

ApplicationsOne common application of inequalities comes from business and involves profit,revenue, and cost. The formula that relates these three quantities is

Increasing the Profit for a Product

The marketing department of a calculator manufacturer has determined that thedemand for a new model of calculator is

Demand equation

where is the price per calculator (in dollars) and represents the number of calculators sold. (If this model is accurate, no one would be willing to pay $100for the calculator. At the other extreme, the company couldn’t sell more than 10million calculators.) The revenue for selling calculators is

Revenue equation

as shown in Figure 2.56. The total cost of producing calculators is $10 percalculator plus a development cost of $2,500,000. So, the total cost is

Cost equation

What price should the company charge per calculator to obtain a profit of at least$190,000,000?

Solution

VerbalModel:

Equation:

To answer the question, solve the inequality

When you write the inequality in general form, find the critical numbers and thetest intervals, and then test a value in each test interval, you can find the solutionto be

as shown in Figure 2.57. Substituting the -values in the original price equationshows that prices of

will yield a profit of at least $190,000,000.


$45.00 ≤ p ≤ $65.00

x

3,500,000 ≤ x ≤ 5,500,000

0.00001x2 90x 2,500,000 ≥ 190,000,000.

P ≥ 190,000,000

P 0.00001x2 90x 2,500,000

P 100x 0.00001x2 10x 2,500,000

P R C

CostRevenueProfit

C 10x 2,500,000.

x

R xp x100 0.00001x

x

xp

0 ≤ x ≤ 10,000,000p 100 0.00001x,

P R C.

CostRevenueProfit

Example 5

332521_0207.qxd 2/8/07 3:00 PM Page 202


Another common application of inequalities is finding the domain of anexpression that involves a square root, as shown in Example 6.

To analyze a test interval, choose a representative -value in the interval and eval-uate the expression at that value. For instance, in Example 6, if you substitute anynumber from the interval into the expression you will obtaina nonnegative number under the radical symbol that simplifies to a real number.If you substitute any number from the intervals and you willobtain a complex number. It might be helpful to draw a visual representation ofthe intervals as shown in Figure 2.59.

4, , 4

64 4x24, 4

x

W RITING ABOUT MATHEMATICS

Profit Analysis Consider the relationship

described on page 202. Write a paragraph discussing why it might be beneficial tosolve if you owned a business. Use the situation described in Example 5 toillustrate your reasoning.

P < 0

P R C

Finding the Domain of an Expression

Find the domain of 64 4x2.

Example 6

Algebraic SolutionRemember that the domain of an expression is the set of all -valuesfor which the expression is defined. Because is defined(has real values) only if is nonnegative, the domain isgiven by


Divide each side by 4.

Write in factored form.

So, the inequality has two critical numbers: and Youcan use these two numbers to test the inequality as follows.

Critical numbers:

Test intervals:

Test: For what values of x is

A test shows that the inequality is satisfied in the closed intervalSo, the domain of the expression is the interval


4, 4.64 4x24, 4.

64 4x2 ≥ 0?

, 4, 4, 4, 4,

x 4, x 4

x 4.x 4

4 x4 x ≥ 0

16 x2 ≥ 0

64 4x2 ≥ 0

64 4x2 ≥ 0.64 4x2

64 4x2x

Graphical SolutionBegin by sketching the graph of the equation

as shown in Figure 2.58. Fromthe graph, you can determine that the -valuesextend from to 4 (including and 4). So,the domain of the expression is theinterval

FIGURE 2.58

−2−4−6 2 4 6−2

2

4

6

10

x

y

64 − 4x2y =

4, 4.64 4x2

44x

y 64 4x2,

−4 4

NonnegativeRadicand

ComplexNumber

ComplexNumber

FIGURE 2.59

332521_0207.qxd 2/8/07 3:00 PM Page 203


Exercises 2.7

In Exercises 1– 4, determine whether each value of is asolution of the inequality.

Inequality Values

1. (a) (b)

(c) (d)

2. (a) (b)

(c) (d)

3. (a) (b)

(c) (d)

4. (a) (b)

(c) (d)

In Exercises 5–8, find the critical numbers of the expression.

5. 6.

7. 8.


9. 10.

11. 12.

13. 14.

15. 16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

In Exercises 27–32, solve the inequality and write thesolution set in interval notation.

27. 28.

29. 30.

31. 32.

Graphical Analysis In Exercises 33–36, use a graphingutility to graph the equation. Use the graph to approximatethe values of that satisfy each inequality.

Equation Inequalities

33. (a) (b)

34. (a) (b)

35. (a) (b)

36. (a) (b)


37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

47.

48.

49.

50.3x

x 1≤

x

x 4 3

5

x 1

2x

x 1< 1

x2 x 6

x≥ 0

x2 2x

x2 9≤ 0

1

x≥

1

x 3

1

x 3≤

9

4x 3

5

x 6>

3

x 2

4

x 5>

1

2x 3

5 7x

1 2x< 4

3x 5

x 5> 4

x 12

x 2 3 ≥ 0

x 6

x 1 2 < 0

1

x 4 < 0

1

x x > 0

y ≥ 36y ≤ 0y x3 x2 16x 16

y ≤ 6y ≥ 0y 18x3

12x

y ≥ 7y ≤ 0y 12x2 2x 1

y ≥ 3y ≤ 0y x2 2x 3

x

x4x 3 ≤ 0x 12x 23 ≥ 0

2x3 x4 ≤ 0x3 4x ≥ 0

4x3 12x2 > 04x3 6x2 < 0

x2 3x 8 > 0

4x2 4x 1 ≤ 0

2x3 13x2 8x 46 ≥ 6

x3 2x2 9x 2 ≥ 20

x3 2x2 4x 8 ≤ 0

x3 3x2 x 3 > 0

2x2 6x 15 ≤ 0

x2 8x 5 ≥ 0

x2 4x 1 > 0

x2 2x 3 < 0

x2 2x > 3x2 x < 6

x2 6x 9 < 16x2 4x 4 ≥ 9

x 32 ≥ 1x 22 < 25

x2 < 36x2 ≤ 9

x

x 2

2

x 12

3

x 5

9x3 25x22x2 x 6

x 3x 0

x 1x 2

x 92x

92

x 4x 5

x 3x 4

x 0x 5x2 x 12 ≥ 0

x 5x 32

x 0x 3x2 3 < 0

x


1. To solve a polynomial inequality, find the ________ numbers of the polynomial, and use these numbersto create ________ ________ for the inequality.

2. The critical numbers of a rational expression are its ________ and its ________ ________.

3. The formula that relates cost, revenue, and profit is ________.

x 2

x 4≥ 3

3x2

x2 4< 1

332521_0207.qxd 2/8/07 3:00 PM Page 204


Graphical Analysis In Exercises 51–54, use a graphingutility to graph the equation. Use the graph to approximatethe values of that satisfy each inequality.

Equation Inequalities

51. (a) (b)

52. (a) (b)

53. (a) (b)

54. (a) (b)

In Exercises 55–60, find the domain of in the expression.Use a graphing utility to verify your result.

55. 56.

57. 58.

59. 60.

In Exercises 61–66, solve the inequality. (Round youranswers to two decimal places.)

61.

62.

63.

64.

65.

66.

67. Height of a Projectile A projectile is fired straightupward from ground level with an initial velocity of 160 feet per second.

(a) At what instant will it be back at ground level?

(b) When will the height exceed 384 feet?

68. Height of a Projectile A projectile is fired straightupward from ground level with an initial velocity of 128 feet per second.

(a) At what instant will it be back at ground level?

(b) When will the height be less than 128 feet?

69. Geometry A rectangular playing field with a perimeter of100 meters is to have an area of at least 500 square meters.Within what bounds must the length of the rectangle lie?

70. Geometry A rectangular parking lot with a perimeter of440 feet is to have an area of at least 8000 square feet.Within what bounds must the length of the rectangle lie?

71. Cost, Revenue, and Profit The revenue and cost equations for a product are

and

where and are measured in dollars and represents thenumber of units sold. How many units must be sold to obtaina profit of at least $750,000? What is the price per unit?

72. Cost, Revenue, and Profit The revenue and cost equations for a product are

and

where and are measured in dollars and represents thenumber of units sold. How many units must be sold to obtaina profit of at least $1,650,000? What is the price per unit?

xCR

C 12x 150,000R x50 0.0002x

xCR

C 30x 250,000R x75 0.0005x

2

3.1x 3.7> 5.8

1

2.3x 5.2> 3.4

1.2x2 4.8x 3.1 < 5.3

0.5x2 12.5x 1.6 > 0

1.3x2 3.78 > 2.12

0.4x2 5.26 < 10.2

x

x2 9 xx2 2x 35

144 9x2x2 7x 12

x2 44 x2

x

y ≤ 0y ≥ 1y 5x

x2 4

y ≤ 2y ≥ 1y 2x2

x2 4

y ≥ 8y ≤ 0y 2x 2

x 1

y ≥ 6y ≤ 0y 3x

x 2

x

73. Cable Television The percents of households in theUnited States that owned a television and had cablefrom 1980 to 2003 can be modeled by

where is the year, with corresponding to 1980.(Source: Nielsen Media Research)

(a) Use a graphing utility to graph the equation.

(b) Complete the table to determine the year in whichthe percent of households that own a television andhave cable will exceed 75%.

(c) Use the trace feature of a graphing utility to verifyyour answer to part (b).

(d) Complete the table to determine the years duringwhich the percent of households that own a televi-sion and have cable will be between 85% and 100%.

(e) Use the trace feature of a graphing utility to verifyyour answer to part (d).

(f ) Explain why the model may give values greater than100% even though such values are not reasonable.

t 0t

0 ≤ t ≤ 23C 0.0031t3 0.216t2 5.54t 19.1,

C

Model It

t 24 26 28 30 32 34

C

t 36 37 38 39 40 41 42 43

C

332521_0207.qxd 2/8/07 3:00 PM Page 205


74. Safe Load The maximum safe load uniformly distributedover a one-foot section of a two-inch-wide wooden beamis approximated by the model where is the depth of the beam.

(a) Evaluate the model for and Use the results to create a bar graph.

(b) Determine the minimum depth of the beam that willsafely support a load of 2000 pounds.

75. Resistors When two resistors of resistances and areconnected in parallel (see figure), the total resistance satisfies the equation

Find for a parallel circuit in which ohms and must be at least 1 ohm.

76. Education The numbers (in thousands) of master’sdegrees earned by women in the United States from 1990to 2002 are approximated by the model

where represents the year, with corresponding to1990 (see figure). (Source: U.S. National Center forEducation Statistics)

(a) According to the model, during what year did the number of master’s degrees earned by women exceed220,000?

(b) Use the graph to verify the result of part (a).

(c) According to the model, during what year will thenumber of master’s degrees earned by women exceed320,000?

(d) Use the graph to verify the result of part (c).

Synthesis


77. The zeros of the polynomial dividethe real number line into four test intervals.

78. The solution set of the inequality is theentire set of real numbers.

Exploration In Exercises 79–82, find the interval for such that the equation has at least one real solution.

79.

80.

81.

82.

83. (a) Write a conjecture about the intervals for inExercises 79–82. Explain your reasoning.

(b) What is the center of each interval for in Exercises79–82?

84. Consider the polynomial and the real numberline shown below.

(a) Identify the points on the line at which the polynomialis zero.

(b) In each of the three subintervals of the line, write thesign of each factor and the sign of the product.

(c) For what -values does the polynomial change signs?

Skills Review

In Exercises 85–88, factor the expression completely.

85.

86.

87.

88.

In Exercises 89 and 90, write an expression for the area ofthe region.

89. 90.

3 + 2b

b

x

1x2 +

2x4 54x

x2x 3 4x 3x 32 16

4x2 20x 25

x

a bx

x ax b

b

b

2x2 bx 5 0

3x2 bx 10 0

x2 bx 4 0

x2 bx 4 0

b

32x2 3x 6 ≥ 0

x3 2x2 11x 12 ≥ 0

t

N

Mas

ter's

deg

rees

ear

ned

(in

thou

sand

s)

Year (0 ↔ 1990)2 4 6 8 10 12 14 16 18

140160180200220240260280300320

t 0t

N 0.03t2 9.6t 172

N

E+

_ R1 R2

RR2 2R1

1

R

1

R1

1

R2

.

RR2R1

d 12.d 10,d 8,d 6,d 4,

dLoad 168.5d 2 472.1,

332521_0207.qxd 2/8/07 3:00 PM Page 206

Chapter Summary 207

Chapter Summary2

What did you learn?

Section 2.1 Review Exercises Analyze graphs of quadratic functions (p. 128). 1, 2

Write quadratic functions in standard form and use the results to 3–18sketch graphs of functions (p. 131).

Use quadratic functions to model and solve real-life problems (p. 133). 19–22

Section 2.2 Use transformations to sketch graphs of polynomial functions (p. 139). 23–28

Use the Leading Coefficient Test to determine the end behavior 29–32of graphs of polynomial functions (p. 141).

Find and use zeros of polynomial functions as sketching aids (p. 142). 33–42

Use the Intermediate Value Theorem to help locate zeros of 43–46polynomial functions (p. 146).

Section 2.3 Use long division to divide polynomials by other polynomials (p. 153). 47–52

Use synthetic division to divide polynomials by binomials of 53–60the form (p. 156).

Use the Remainder Theorem and the Factor Theorem (p. 157). 61–64

Section 2.4 Use the imaginary unit to write complex numbers (p. 162). 65–68

Add, subtract, and multiply complex numbers (p. 163). 69–74

Use complex conjugates to write the quotient of two complex numbers 75–78in standard form (p. 165).

Find complex solutions of quadratic equations (p. 166). 79–82

Section 2.5 Use the Fundamental Theorem of Algebra to determine the number of 83–88

zeros of polynomial functions (p. 169).

Find rational zeros of polynomial functions (p. 170). 89–96

Find conjugate pairs of complex zeros (p. 173). 97, 98

Use factoring (p. 173), Descartes’s Rule of Signs (p. 176), and the Upper and 99–110Lower Bound Rules (p. 177), to find zeros of polynomials.

Section 2.6 Find the domains of rational functions (p. 184). 111–114

Find the horizontal and vertical asymptotes of graphs of rational functions (p. 185). 115–118

Analyze and sketch graphs of rational functions (p. 187). 119–130

Sketch graphs of rational functions that have slant asymptotes (p. 190). 131–134

Use rational functions to model and solve real-life problems (p. 191). 135–138

Section 2.7 Solve polynomial inequalities (p. 197), and rational inequalities (p. 201). 139–146

Use inequalities to model and solve real-life problems (p. 202). 147, 148

i

x k

332521_020R.qxd 2/8/07 3:01 PM Page 207

In Exercises 1 and 2, graph each function. Comparethe graph of each function with the graph of

1. (a)

(b)

(c)

(d)

2. (a)

(b)

(c)

(d)

In Exercises 3–14, write the quadratic function in standardform and sketch its graph. Identify the vertex, axis ofsymmetry, and -intercept(s).

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

In Exercises 15–18, write the standard form of the equationof the parabola that has the indicated vertex and whosegraph passes through the given point.

15. 16.

17. Vertex: point:

18. Vertex: point:

19. Geometry The perimeter of a rectangle is 200 meters.

(a) Draw a diagram that gives a visual representation ofthe problem. Label the length and width as and respectively.

(b) Write as a function of Use the result to write thearea as a function of

(c) Of all possible rectangles with perimeters of 200meters, find the dimensions of the one with the maxi-mum area.

20. Maximum Revenue The total revenue earned (indollars) from producing a gift box of candles is given by

where is the price per unit (in dollars).

(a) Find the revenues when the prices per box are $20,$25, and $30.

(b) Find the unit price that will yield a maximum revenue.What is the maximum revenue? Explain your results.

21. Minimum Cost A soft-drink manufacturer has dailyproduction costs of

where is the total cost (in dollars) and is the number ofunits produced. How many units should be produced eachday to yield a minimum cost?

22. Sociology The average age of the groom at a firstmarriage for a given age of the bride can be approximatedby the model

where is the age of the groom and is the age of thebride. Sketch a graph of the model. For what age of thebride is the average age of the groom 26? (Source: U.S.Census Bureau)

In Exercises 23–28, sketch the graphs of andthe transformation.

23.

24.

25.

26.

27.

28. f x 12x5 3y x5,

f x x 35y x5,

f x 2x 24y x 4,

f x 2 x 4y x4,

f x 4x3y x3,

f x x 43y x3,

y x n2.2

xy

20 ≤ x ≤ 25y 0.107x2 5.68x 48.5,

xC

C 70,000 120x 0.055x 2

p

Rp 10p2 800p

R

x.x.y

y,x

1, 62, 3;2, 31, 4;

x−2 2 4 6

2

6

(2, 2)

(0, 3)

y

x4 8

2

−2

−4

−6

(4, 1)

(2, −1)

y

f x 126x2 24x 22

f x 13x2 5x 4

f x 4x 2 4x 5

hx x2 5x 4

f x x2 6x 1

hx 4x2 4x 13

f x x2 8x 12

f t 2t 2 4t 1

hx 3 4x x2

f x x2 8x 10

f x 6x x2

gx x2 2x

x

kx 12x 2 1

hx x 32

gx 4 x 2

f x x 2 4

kx x 22

hx x 2 2

gx 2x 2

f x 2x 2

y x2.2.1


Review Exercises2

332521_020R.qxd 2/8/07 3:01 PM Page 208

Review Exercises 209

In Exercises 29–32, describe the right-hand and left-handbehavior of the graph of the polynomial function.

29.

30.

31.

32.

In Exercises 33–38, find all the real zeros of the polynomialfunction. Determine the multiplicity of each zero and thenumber of turning points of the graph of the function. Usea graphing utility to verify your answers.

33. 34.

35. 36.

37. 38.

In Exercises 39– 42, sketch the graph of the function by (a)applying the Leading Coefficient Test, (b) finding the zerosof the polynomial, (c) plotting sufficient solution points,and (d) drawing a continuous curve through the points.

39.

40.

41.

42.

In Exercises 43– 46, (a) use the Intermediate Value Theoremand the table feature of a graphing utility to find intervalsone unit in length in which the polynomial function isguaranteed to have a zero. (b) Adjust the table to approxi-mate the zeros of the function. Use the zero or root featureof the graphing utility to verify your results.

43.

44.

45.

46.

In Exercises 47–52, use long division to divide.

47.

48.

49.

50.

51.

52.

In Exercises 53–56, use synthetic division to divide.

53.

54.

55.

56.

In Exercises 57 and 58, use synthetic division to determinewhether the given values of are zeros of the function.

57.

(a) (b) (c) (d)

58.

(a) (b) (c) (d)

In Exercises 59 and 60, use synthetic division to find eachfunction value.

59.

(a) (b)

60.

(a) (b)

In Exercises 61– 64, (a) verify the given factor(s) of the func-tion (b) find the remaining factors of (c) use your resultsto write the complete factorization of (d) list all real zerosof and (e) confirm your results by using a graphing utilityto graph the function.

Function Factor(s)

61.

62.

63.

64.

In Exercises 65– 68, write the complex number instandard form.

65. 66.

67. 68.


69.

70.

71. 72.

73. 74. i6 i3 2i10 8i2 3i 1 6i5 2i 5i13 8i

2

2

2

2i 2

2

2

2i

7 5i 4 2i

5i i2i2 3i

3 256 4

2.4

x 2x 5f x x4 11x3 41x2 61x 30

x 2x 3f x x 4 4x 3 7x2 22x 24

x 6f x 2x3 11x2 21x 90

x 4f x x3 4x2 25x 28

f,f,f,f,

g2 g4gt 2t 5 5t 4 8t 20

f 1f 3f x x4 10x3 24x 2 20x 44

x 1x 23x 4x 4

f x 3x3 8x2 20x 16

x 1x 0x 34x 1

f x 20x4 9x3 14x 2 3x

x

3x3 20x2 29x 12x 3

2x3 19x 2 38x 24

x 4

0.1x3 0.3x 2 0.5

x 5

6x4 4x3 27x 2 18x

x 2

6x4 10x3 13x 2 5x 2

2x2 1

x4 3x3 4x 2 6x 3

x2 2

3x4

x 2 1

5x 3 13x 2 x 2

x2 3x 1

4x 7

3x 2

24x 2 x 8

3x 2

2.3

f x 7x 4 3x 3 8x2 2

f x x 4 5x 1

f x 0.25x 3 3.65x 6.12

f x 3x 3 x2 3

hx 3x2 x4

f x xx3 x2 5x 3gx 2x3 4x2

f x x3 x2 2

gx x4 x3 2x2f x 12x3 20x2

f x x3 8x2f t t 3 3t

f x xx 32f x 2x2 11x 21

hx x5 7x 2 10x

gx 34x4 3x 2 2

f x 12 x3 2x

f x x 2 6x 9

332521_020R.qxd 2/8/07 3:01 PM Page 209

In Exercises 75 and 76, write the quotient in standard form.

75. 76.

In Exercises 77 and 78, perform the operation and write theresult in standard form.

77. 78.

In Exercises 79– 82, find all solutions of the equation.

79. 80.

81. 82.

In Exercises 83–88, find all the zeros of the function.

83.

84.

85.

86.

87.

88.

In Exercises 89 and 90, use the Rational Zero Test to list allpossible rational zeros of f.

89.

90.

In Exercises 91–96, find all the rational zeros of the function.

91.

92.

93.

94.

95.

96.

In Exercises 97 and 98, find a polynomial function with realcoefficients that has the given zeros. (There are manycorrect answers.)

97. 98.

In Exercises 99–102, use the given zero to find all the zerosof the function.

Function Zero

99.

100.

101.

102.

In Exercises 103–106, find all the zeros of the function andwrite the polynomial as a product of linear factors.

103.

104.

105.

106.

In Exercises 107 and 108, use Descartes’s Rule of Signs todetermine the possible numbers of positive and negativezeros of the function.

107.

108.

In Exercises 109 and 110, use synthetic division to verifythe upper and lower bounds of the real zeros of

109.

(a) Upper:

(b) Lower:

110.

(a) Upper:

(b) Lower:

In Exercises 111–114, find the domain of the rationalfunction.

111. 112.

113. 114.

In Exercises 115–118, identify any horizontal or verticalasymptotes.

115. 116.

117. 118.

In Exercises 119–130, (a) state the domain of the function,(b) identify all intercepts, (c) find any vertical and horizontalasymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function.

119. 120.

121. 122.

123. 124.

125. 126. hx 4

x 12f x

x

x 2 1

f x 2x

x 2 4px

x 2

x 2 1

hx x 3

x 2gx

2 x

1 x

f x 4

xf x

5

x 2

hx x3 4x2

x2 3x 2hx

2x 10x2 2x 15

f x 2x2 5x 3

x2 2f x

4x 3

f x x2 x 2

x2 4f x

8x2 10x 24

f x 3x2

1 3xf x

5xx 12

2.6

x 4

x 8

f x 2x3 5x2 14x 8

x 14

x 1

f x 4x3 3x2 4x 3

f.

hx 2x5 4x3 2x 2 5

gx 5x3 3x 2 6x 9

f x x4 8x3 8x2 72x 153

gx x4 4x3 3x2 40x 208

gx x3 7x2 36

f x x3 4x2 5x

1 if x 4x4 11x3 14x2 6x

2 igx 2x 4 3x 3 13x2 37x 15

4ih x x3 2x2 16x 32

if x x3 4x2 x 4

1 2i3,2,3 i4,23,

f x 25x4 25x3 154x2 4x 24

f x x 4 x3 11x2 x 12

f x x3 9x2 24x 20

f x x3 10x2 17x 8

f x 3x3 20x2 7x 30

f x x3 2x2 21x 18

f x 3x4 4x3 5x 2 8

f x 4x3 8x 2 3x 15

f x x 8x 52x 3 ix 3 if x x 4x 6x 2ix 2if x x 3 6x

f x x2 9x 8

f x x 4x 92

f x 3xx 22

2.5

6x2 3x 27 0x2 2x 10 0

2 8x2 03x2 1 0

1

2 i

5

1 4i

4

2 3i

2

1 i

3 2i

5 i

6 i

4 i


332521_020R.qxd 2/8/07 3:01 PM Page 210

Review Exercises 211

127. 128.

129. 130.

In Exercises 131–134, (a) state the domain of the function,(b) identify all intercepts, (c) identify any vertical and slantasymptotes, and (d) plot additional solution points asneeded to sketch the graph of the rational function.

131. 132.

133.

134.

135. Average Cost A business has a production cost offor producing units of a product. The

average cost per unit, is given by

Determine the average cost per unit as increases withoutbound. (Find the horizontal asymptote.)

136. Seizure of Illegal Drugs The cost (in millions ofdollars) for the federal government to seize of anillegal drug as it enters the country is given by


(b) Find the costs of seizing 25%, 50%, and 75% of thedrug.

(c) According to this model, would it be possible to seize100% of the drug?

137. Page Design A page that is inches wide and incheshigh contains 30 square inches of print. The top andbottom margins are 2 inches deep and the margins oneach side are 2 inches wide.

(a) Draw a diagram that gives a visual representation ofthe problem.

(b) Show that the total area on the page is

(c) Determine the domain of the function based on thephysical constraints of the problem.

(d) Use a graphing utility to graph the area function andapproximate the page size for which the least amountof paper will be used. Verify your answer numericallyusing the table feature of the graphing utility.

138. Photosynthesis The amount of uptake (inmilligrams per square decimeter per hour) at optimaltemperatures and with the natural supply of isapproximated by the model

where is the light intensity (in watts per square meter).Use a graphing utility to graph the function and determinethe limiting amount of uptake.

In Exercises 139–146, solve the inequality.

139. 140.

141. 142.

143. 144.

145. 146.

147. Investment dollars invested at interest rate compounded annually increases to an amount

in 2 years. An investment of $5000 is to increase to anamount greater than $5500 in 2 years. The interest ratemust be greater than what percent?

148. Population of a Species A biologist introduces 200ladybugs into a crop field. The population of theladybugs is approximated by the model

where is the time in days. Find the time required for thepopulation to increase to at least 2000 ladybugs.

Synthesis

True or False? In Exercises 149 and 150, determinewhether the statement is true or false. Justify your answer.

149. A fourth-degree polynomial with real coefficients canhave and 5 as its zeros.

150. The domain of a rational function can never be the set ofall real numbers.

151. Writing Explain how to determine the maximum orminimum value of a quadratic function.

152. Writing Explain the connections among factors of apolynomial, zeros of a polynomial function, and solutionsof a polynomial equation.

153. Writing Describe what is meant by an asymptote of agraph.

4i,8i,5,

t

P 10001 3t

5 t

P

A P1 r2

rP

1x 2

>1x

x2 7x 12x

≥ 0

x 53 x

< 02

x 1≤

3x 1

12x3 20x2 < 0x3 16x ≥ 0

2x2 x ≥ 156x2 5x < 4

2.7

CO2

x

x > 0y 18.47x 2.96

0.23x 1,

CO2

CO2y

A 2x2x 7

x 4.

A

yx

0 ≤ p < 100.C 528p

100 p,

p%C

x

x > 0.C C

x

0.5x 500

x,

C,xC 0.5x 500

f x 3x3 4x2 12x 16

3x2 5x 2

f x 3x3 2x2 3x 2

3x2 x 4

f x x2 1x 1

f x 2x3

x2 1

f x 6x2 7x 2

4x2 1f x

6x2 11x 33x2 x

y 2x 2

x 2 4f x

6x2

x 2 1

332521_020R.qxd 2/8/07 3:01 PM Page 211


Chapter Test2

Take this test as you would take a test in class. When you are finished, check yourwork against the answers given in the back of the book.

1. Describe how the graph of differs from the graph of

(a) (b)

2. Find an equation of the parabola shown in the figure at the left.

3. The path of a ball is given by where is the height (in feet) ofthe ball and is the horizontal distance (in feet) from where the ball was thrown.

(a) Find the maximum height of the ball.

(b) Which number determines the height at which the ball was thrown? Doeschanging this value change the coordinates of the maximum height of the ball?Explain.

4. Determine the right-hand and left-hand behavior of the graph of the functionThen sketch its graph.

5. Divide using long division. 6. Divide using synthetic division.

7. Use synthetic division to show that is a zero of the function given by

Use the result to factor the polynomial function completely and list all the real zerosof the function.

8. Perform each operation and write the result in standard form.

(a) (b)

9. Write the quotient in standard form:

In Exercises 10 and 11, find a polynomial function with real coefficients that has thegiven zeros. (There are many correct answers.)

10. 11.

In Exercises 12 and 13, find all the zeros of the function.

12. 13.

In Exercises 14–16, identify any intercepts and asymptotes of the graph the function.Then sketch a graph of the function.

14. 15. 16.

In Exercises 17 and 18, solve the inequality. Sketch the solution set on the realnumber line.

17. 18.2

x>

5

x 62x2 5x > 12

gx x 2 2

x 1f x

2x2 5x 12x2 16

hx 4

x 2 1

f x x4 9x2 22x 24f x x3 2x2 5x 10

1 3 i, 1 3 i, 2, 20, 3, 3 i, 3 i

5

2 i.

2 3 i2 3 i10i 3 25

f x 4x3 x 2 12x 3.

x 3

2x4 5x 2 3

x 2

3x 3 4x 1

x 2 1

h t 34t 5 2t 2.

xyy

120 x 2 3x 5,

gx x 322

gx 2 x 2

f x x 2.g

x−2−4

−4

2 4 6 8

2

4

6

−6

(0, 3)

(3, −6)

y

FIGURE FOR 2

332521_020R.qxd 2/8/07 3:01 PM Page 212

213

Proofs in MathematicsThese two pages contain proofs of four important theorems about polynomialfunctions. The first two theorems are from Section 2.3, and the second twotheorems are from Section 2.5.

ProofFrom the Division Algorithm, you have

and because either or the degree of is less than the degree of you know that must be a constant. That is, Now, by evaluating at you have

To be successful in algebra, it is important that you understand the connec-tion among factors of a polynomial, zeros of a polynomial function, and solutionsor roots of a polynomial equation. The Factor Theorem is the basis for thisconnection.

ProofUsing the Division Algorithm with the factor you have

By the Remainder Theorem, and you have

where is a polynomial of lesser degree than If then

and you see that is a factor of Conversely, if is a factor ofdivision of by yields a remainder of 0. So, by the Remainder

Theorem, you have f k 0.x kf xf x,

x kf x.x k

f x x kqx

f k 0,f x.qx

f x x kqx fk

rx r f k,

f x x kqx rx.

x k,

0qk r r.

f k k kqk r

x k,f xrx r.rx

x k,rxrx 0

f x x kqx rx

The Remainder Theorem (p. 157)

If a polynomial is divided by the remainder is

r f k.

x k,f x

The Factor Theorem (p. 157)

A polynomial has a factor if and only if f k 0.x kf x

332521_020R.qxd 2/8/07 3:01 PM Page 213

214

Proofs in Mathematics

ProofUsing the Fundamental Theorem of Algebra, you know that must have at leastone zero, Consequently, is a factor of and you have

If the degree of is greater than zero, you again apply the FundamentalTheorem to conclude that must have a zero which implies that

It is clear that the degree of is that the degree of is andthat you can repeatedly apply the Fundamental Theorem times until you obtain

where is the leading coefficient of the polynomial

ProofTo begin, you use the Linear Factorization Theorem to conclude that can becompletely factored in the form

If each is real, there is nothing more to prove. If any is complex then, because the coefficients of are real, you know that the conju-

gate is also a zero. By multiplying the corresponding factors, youobtain

where each coefficient is real.

x2 2ax a2 b2

x cix cj x a bix a bi

cj a bif xb 0,

a bi,ci cici

f x dx c1x c2x c3 . . . x cn.

f x

f x.an


nn 2,f2xn 1,f1x

f x x c1x c2f2x.

c2,f1

f1x

f x x c1f1x.

f x,x c1c1.f

The FundamentalTheorem of Algebra

The Linear FactorizationTheorem is closely related tothe Fundamental Theorem ofAlgebra. The FundamentalTheorem of Algebra has a longand interesting history. In theearly work with polynomialequations, The FundamentalTheorem of Algebra wasthought to have been not true,because imaginary solutionswere not considered. In fact,in the very early work bymathematicians such asAbu al-Khwarizmi (c. 800 A.D.),negative solutions were also notconsidered.

Once imaginary numberswere accepted, several mathe-maticians attempted to give ageneral proof of theFundamental Theorem ofAlgebra. These includedGottfried von Leibniz (1702),Jean d’Alembert (1746),Leonhard Euler (1749), Joseph-Louis Lagrange (1772), andPierre Simon Laplace (1795).The mathematician usuallycredited with the first correctproof of the FundamentalTheorem of Algebra is CarlFriedrich Gauss, who publishedthe proof in his doctoral thesisin 1799.

Linear Factorization Theorem (p. 169)

If is a polynomial of degree where then has precisely linear factors

where are complex numbers.c1, c2, . . . , cn


nfn > 0,n,f x

Factors of a Polynomial (p. 173)

Every polynomial of degree with real coefficients can be written asthe product of linear and quadratic factors with real coefficients, where thequadratic factors have no real zeros.

n > 0

332521_020R.qxd 2/8/07 3:01 PM Page 214

215

1. Show that if then where using long division. Inother words, verify the Remainder Theorem for a third-degree polynomial function.

2. In 2000 B.C., the Babylonians solved polynomial equationsby referring to tables of values. One such table gave the val-ues of To be able to use this table, the Babylonianssometimes had to manipulate the equation as shown below.

Original equation

Multiply each side by

Rewrite.

Then they would find in the column of thetable. Because they knew that the corresponding -value wasequal to they could conclude that

(a) Calculate for 2, 3, . . . , 10. Record thevalues in a table.

Use the table from part (a) and the method above to solveeach equation.

(b)

(c)

(d)

(e)

(f)

(g)

Using the methods from this chapter, verify your solution toeach equation.

3. At a glassware factory, molten cobalt glass is poured intomolds to make paperweights. Each mold is a rectangularprism whose height is 3 inches greater than the length of eachside of the square base. A machine pours 20 cubic inches of liquid glass into each mold. What are the dimensions ofthe mold?

4. Determine whether the statement is true or false. If false,provide one or more reasons why the statement is false andcorrect the statement. Let

and let Then

where is a second-degree polynomial.

5. The parabola shown in the figure has an equation of theform Find the equation of this parabolaby the following methods. (a) Find the equation analytically.(b) Use the regression feature of a graphing utility to findthe equation.

6. One of the fundamental themes of calculus is to find theslope of the tangent line to a curve at a point. To see how thiscan be done, consider the point on the graph of thequadratic function

(a) Find the slope of the line joining and Is theslope of the tangent line at greater than or lessthan the slope of the line through and

(b) Find the slope of the line joining and Is theslope of the tangent line at greater than or lessthan the slope of the line through and

(c) Find the slope of the line joining and Is the slope of the tangent line at greater than orless than the slope of the line through and

(d) Find the slope of the line joining andin terms of the nonzero number

(e) Evaluate the slope formula from part (d) for 1, and 0.1. Compare these values with those in parts(a)–(c).

(f ) What can you conclude the slope of the tangent line atto be? Explain your answer.2, 4

h 1,

h.2 h, f 2 h2, 4

2.1, 4.41?2, 4

2, 42.1, 4.41.2, 41, 1?2, 4

2, 41, 1.2, 4

3, 9?2, 42, 4

3, 9.2, 4

x31 2−3 −1−2

1

2

3

4

5(2, 4)

y

f x x2.2, 4

(0, −4)

(6, −10)

(1, 0)

(2, 2)(4, 0)

−4

2

−6

6−2−4 8x

y

y ax2 bx c.

qx

f xx 1

qx 2

x 1

f 2 1.a 0,f x ax3 bx2 cx d,

10x3 3x2 297

7x3 6x2 1728

2x3 5x2 2500

3x3 x2 90

x3 2x2 288

x3 x2 252

y 1,y3 y2

x bya.axb,y

y3 y2a2cb3

axb

3

axb

2

a2 cb3

a2

b3. a3 x3

b3 a2 x2

b2 a2 cb3

ax3 bx2 c

y3 y2.

r ak3 bk2 ck df k r,f x ax3 bx2 cx d

P.S. Problem Solving

This collection of thought-provoking and challenging exercises further exploresand expands upon concepts learned in this chapter.

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7. Use the form to create a cubicfunction that (a) passes through the point and rises tothe right and (b) passes through the point and fallsto the right. (There are many correct answers.)

8. The multiplicative inverse of is a complex number such that Find the multiplicative inverse ofeach complex number.

(a) (b) (c)

9. Prove that the product of a complex number and itscomplex conjugate is a real number.

10. Match the graph of the rational function given by

with the given conditions.

(a) (b)

(c) (d)

(i) (ii) (iii) (iv)

11. Consider the function given by

(a) Determine the effect on the graph of if and is varied. Consider cases in which is positive and isnegative.

(b) Determine the effect on the graph of if and is varied.

12. The endpoints of the interval over which distinct vision ispossible is called the near point and far point of the eye(see figure). With increasing age, these points normallychange. The table shows the approximate near points (ininches) for various ages (in years).

FIGURE FOR 12

(a) Use the regression feature of a graphing utility to finda quadratic model for the data. Use a graphing utility toplot the data and graph the model in the same viewingwindow.

(b) Find a rational model for the data. Take the reciprocalsof the near points to generate the points Usethe regression feature of a graphing utility to find a lin-ear model for the data. The resulting line has the form

Solve for Use a graphing utility to plot the data andgraph the model in the same viewing window.

(c) Use the table feature of a graphing utility to create atable showing the predicted near point based on eachmodel for each of the ages in the original table. Howwell do the models fit the original data?

(d) Use both models to estimate the near point for a personwho is 25 years old. Which model is a better fit?

(e) Do you think either model can be used to predict thenear point for a person who is 70 years old? Explain.

y.

1y

ax b.

x, 1y.

Objectblurry

Objectblurry

Objectclear

Nearpoint

Farpoint

xy

ba 0f

aaab 0f

f x ax

x b2.

d > 0d < 0d < 0d < 0

c > 0c > 0c < 0c > 0

b < 0b > 0b > 0b < 0

a > 0a < 0a > 0a > 0

x

y

x

y

x

y

x

y

f x ax bcx d

a bi

z 2 8iz 3 iz 1 i

z zm 1.zmz

3, 12, 5

f x x kqx r

216

Age, x Near point, y

16 3.0

32 4.7

44 9.8

50 19.7

60 39.4

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