Stewart J. College Algebra: Concepts and Contexts

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504 CHAPTER 6

■

Power, Polynomial, and Rational Functions

■ Polynomial Functions

A polynomial function is a function that is a sum of power functions. The following

are examples of polynomial functions:

Recall that the degree of a polynomial is the highest power of the variable that ap-

pears in the polynomial. So the degrees of the above polynomials are 3, 4, and 1, re-

spectively. In general, we have the following definition.

Polynomial Functions

f 1x2= 3x

+ 2x - 1g1x 2= x

+ 2x

h1x 2= 3x + 2

One way to graph a polynomial function is to graph the terms of the polynomial

(they are power functions), then use graphical addition.

A polynomial function is a function of the form

where are real numbers and n is a nonnegative integer.

■

If , then the polynomial has degree n.

■

The term is the leading term of the polynomial.a

⫽ 0

, a

, . . . , a

P1x 2= a

+ a

n -1

+ a

x + a

6.3 Polynomial Functions: Combining Power Functions

■

Polynomial Functions

■

Graphing Polynomial Functions by Factoring

■

End Behavior and the Leading Term

■

Modeling with Polynomial Functions

IN THIS SECTION… we study polynomial functions as sums of power functions. We

learn how to graph a polynomial function using factoring.

GET READY… by reviewing how to factor polynomial expressions in Algebra Toolkit

B.2. Test your understanding by doing the Algebra Checkpoint at the end of this section.

Polynomial functions are obtained by combining power functions using the al-

gebraic operations on functions we studied in Section 6.1. In this section we’ll

encounter an application of polynomial functions to the ordinary task of mailing

a package at the post office. The U.S. Postal Service has restrictions on the size

of packages that it will mail. The rule is: The length of the package plus the dis-

tance around the package must not exceed a specified number of inches. To find

out the dimensions of the largest box that the Postal Service will mail requires

using polynomial functions, as we’ll see in Example 6. But first we must learn

how to graph polynomial functions.

Polynomial functions

at the post office

pzAxe/Shutterstock.com 2009

SECTION 6.3

■

Polynomial Functions: Combining Power Functions 505

example

Graphing a Polynomial with Graphical Addition

Graph the polynomial function by expressing it as a sum of

power functions and then using graphical addition.

Solution

The polynomial function f is the sum of the three power functions

We first graph and ; then we use graphical addition to graph their

sum , as in Figure 1(a). We now shift this graph up 1 unit to obtain the

graph of , as in Figure 1(b).y = x

- 3x + 1

y = x

- 3x

y =-3xy = x

y = x





y =-3xy = 1

f 1x2= x

- 3x + 1

1_3 2 3

(

)

Gra

h of

=x£-3x

y=x£

y=_3x

1_1_3 2 3

(

)

Gra

h of

=x£-3x+1

y=x£-3x

y=x£-3x+1

figure 1 Graphing a polynomial function with graphical addition

■ NOW TRY EXERCISE 7 ■

■ Graphing Polynomial Functions by Factoring

If a polynomial function can be factored into linear factors, then an effective way to

graph the polynomial is to first use the zero-product property to find the x-intercepts.

Between consecutive x-intercepts the polynomial must be always positive or always

negative; to decide which sign is the correct one, we use test points. The process is

illustrated in the next example.

Intercepts are reviewed in

Algebra Toolkit D.2, page T71.

example

Graphing a Polynomial Function

Let .

(a) Show that .

(b) Find the x-intercepts of the graph of f.

(d) Sketch a graph of f.

f 1x2= 1x - 321x + 221x - 22

f 1x2= x

- 3x

- 4x + 12

506 CHAPTER 6

■

Power, Polynomial, and Rational Functions

(d) We first plot the x-intercepts as well as the other points calculated in Figure 2.

The graph of f is above the x-axis on intervals where f is positive and below

the x-axis where f is negative. From Figure 2 we know the sign of f on each

interval, so we can complete the graph as in Figure 3.

■ NOW TRY EXERCISE 9 ■

Solution

(a) We expand the product to see whether it is the same expression as the expres-

sion defining f.

Sum and difference of terms

Distributive Property

This last expression is the same as the expression defining f.

(b) The x-intercepts are the solutions to the equation . We have

Set equal to 0

Factored form

Zero-Product Property

Solve

So the x-intercepts are 3, , and 2.

The function f is either positive or negative on each of these intervals. To

decide which it is, we pick a test point in the interval and evaluate f at that

point. For example, in the interval , let’s pick the test point .

Evaluating f at , we get

Factored form of f

Evaluate f at

Calculate

Since is less than 0, f is negative at each point in the entire interval

. In Figure 2 we choose test points in each of the four intervals and

determine the sign of the function f on each interval.

1- q, - 22

- 30

= 1- 621- 121- 52=-30

- 3 f 1- 32= 1- 3 - 321- 3 + 221- 3 - 2 2

f 1x 2= 1x - 3 21x + 221x - 22

- 3

- 31- q, - 22

1- q, - 22,1- 2, 22,12, 32,and13, q 2

- 2

x = 3 o r  x =-2 o r  x = 2

x - 3 = 0 orx + 2 = 0 orx - 2 = 0

1x - 321x + 221x - 22= 0

f 1x 2 x

- 3x

- 4x + 12 = 0

f 1x2= 0

= x

- 3x

- 4x + 12

= x

1x - 3 2- 41x - 32

1x - 321x + 221x - 22= 1x - 321x

- 42

It is usually easier to evaluate a

polynomial by using the factored

form. In Example 1 we used the

factored form to evaluate f at .- 3

432 2.50_2_3x

12_1.12512_30f(x)

+_+_Sign

Test point Test point Test point

Test point

figure 2 Intercepts and test points for f 1x 2= x

- 3x

- 4x + 12

figure 3 Graph of

f 1x 2= x

- 3x

- 4x + 12

SECTION 6.3

■

Polynomial Functions: Combining Power Functions 507

example

Graphing a Polynomial Function

Let .

(a) Express the function f in factored form.

(b) Find the x-intercepts of the graph of f.

(d) Sketch a graph of f.

Solution

(a) We factor the polynomial completely.

Definition of f

Factor

Factor trinomial

(b) The x-intercepts are the solutions to the equation . We use the

factored form of f to solve this equation.

Set equal to 0

Zero-Product Property

Solve

So the x-intercepts are 0, , and 1.

To determine the sign of f on each of these intervals, we use test points as

shown in the diagram in Figure 4.

A- q, -

B,A-

, 0B,10, 1 2,and11, q 2

x = 0 or x =-



or x = 1

- x

= 0 or 2 x + 3 = 0 or x - 1 = 0

f 1x 2 - x

12x + 321x - 12= 0

f 1x2= 0

=-x

12x + 3 21x - 1 2

- x

=-x

12x

+ x - 32

f 1x 2=-2x

- x

+ 3x

f 1x2=-2x

- x

+ 3x

1.50.50_1.5 1_2 _1x

_6.750.5_12f(x)

_+_

+Sign

Test pointTest pointTest pointTest point

figure 4 Intercepts and test points for f 1x 2=-2x

- x

+ 3x

(d) We plot the x-intercepts as well as the test points that we calculated in the

diagram in Figure 4. The graph of f is above the x-axis on intervals where f is

positive and below the x-axis where f is negative, as shown in Figure 5.

■ NOW TRY EXERCISE 19 ■

_12

figure 5 Graph of

f 1x 2=-2x

- x

+ 3x

example

Graphing a Polynomial Function

Let .

(a) Express the function f in factored form.

(b) Find the x-intercepts of the graph of f.

f 1x2= x

- 2x

- 4x + 8

508 CHAPTER 6

■

Power, Polynomial, and Rational Functions

30_2 2_3x

58_25f(x)

++_Sign

Test pointTest pointTest point

figure 6 Intercepts and test points for

f 1x 2= x

- 2x

- 4x + 8

(d) Sketch a graph of f.

Solution

(a) We factor the polynomial by grouping.

Definition of f

Group

Factor each term

Factor from each term

Factor difference of squares

(b) The x-intercepts are the solutions to the equation . We use the

factored form of f to solve this equation.

Set equal to 0

Zero-Product Property

Solve

So the x-intercepts are and 2.

To determine the sign of f on each of these intervals, we use test points as

shown in the diagram in Figure 6.

1- q, - 22,

1- 2, 22,and12, q 2

- 2

x =-2 or x = 2 or x = 2

x + 2 = 0 or x - 2 = 0 or x - 2 = 0

f 1x 2 1x + 2 21x - 2 21x - 22= 0

f 1x2= 0

= 1x + 221x - 221x - 2 2

x - 2 = 1x

- 421x - 22

= x

1x - 2 2- 41x - 22

= 1x

- 2x

2+ 1- 4x + 8 2

f 1x 2= x

- 2x

- 4x + 8

(d) We plot the x-intercepts as well as the test points that we calculated in the

diagram in Figure 6. The graph of f is above the x-axis on intervals where f is

positive and below the x-axis where f is negative, as shown in Figure 7.

■ NOW TRY EXERCISE 23 ■

figure 7 Graph of

f 1x 2= x

- 2x

- 4x + 8

■ End Behavior and the Leading Term

The end behavior of a polynomial function is a description of what happens to the

values of the function as x becomes large in the positive or negative direction. To ex-

plain end behavior, we use the following arrow notation.

SECTION 6.3

■

Polynomial Functions: Combining Power Functions 509

means “x becomes large in the positive direction”

means “x becomes large in the negative direction”x S

- q

x Sq

Arrow Notation

For example, the power functions and have the end behavior shown in

Figure 8.

y = x

y=x£

y=x™

_10

_20

_30

_10

_2_4 42_2_3 _1 213

as x _`

` y

as x `

figure 8 End behavior of and y = x

y = x

Power functions with even powers have end behavior like that of or

, and power functions with odd powers have end behavior like that of

or . But how do we determine the end behavior of any polynomial? For any

polynomial function the end behavior is completely determined by the leading term.

End Behavior

y =-x

y = x

y =-x

y = x

A polynomial function has the same end behavior as the power function

consisting of its leading term.

So the polynomial function has the

same end behavior as the power function . The reason is that for large

values of x the other terms are relatively insignificant in size when compared to the

leading term. The next example illustrates this relationship.

Q1x 2= a

P1x 2= a

+ a

n -1

+ a

x + a

example

Graphing a Polynomial Function

Let .

(a) Determine the end behavior of the leading term of P.

(b) Confirm that P and its leading term have the same end behavior by comparing

their values for large values of x.

P1x 2= 3x

- 5x

+ 2x

510 CHAPTER 6

■

Power, Polynomial, and Rational Functions

them together.

Solution

(a) The leading term of P is the function . Since Q is a power function

with odd power, its end behavior is as follows:

(b) We make a table of values of P and Q for large values of x. We see that when x

is large, P and Q have approximately the same value (in each case ).

This confirms that they have the same end behavior as .x Sq

y Sq

y Sq as x Sq

andy S - q as x S - q

Q1x 2= 3x

P1x 2ⴝ 3x

ⴚ 5x

ⴙ 2x Q1x2ⴝ 3x

15 2,261,280 2,278,125

30 72,765,060 72,900,000

50 936,875,100 937,500,000

rectangles. The larger the viewing rectangle, the more the graphs look alike.

This confirms that they have the same end behavior.

10,000

_10,000

_10 10

_50

_3 3

_2 2

_1 1

figure 9 Graphs of P and Q in progressively larger viewing rectangles

■ NOW TRY EXERCISE 27 ■

■ Modeling with Polynomial Functions

We have seen that the graphs of polynomial functions often have local maximum and

minimum values. In the applications of polynomials these points often play an im-

portant role, as we’ll see in the next example.

Local maxima and minima of

functions are studied in Section 1.7.

example

Finding Maximum Volume

Your local post office will mail a package only if its length plus girth is 108 inches

(or less). April wants to mail a package in the shape of a rectangular box with a

square base. To get the largest volume, she wants the length plus the girth to be ex-

actly 108 inches.

(a) Express the volume of the package as a function of the side x of the base.

(b) Use a graphing calculator to graph in the viewing rectangle [0, 30] by

[0, 12,000]. For what value of x does the maximum volume occur?

at the post office?

SECTION 6.3

■

Polynomial Functions: Combining Power Functions 511

figure 10 Girth is the “distance

around.”

Solution

(a) Let x be the side of the base and let L be the length, as shown in Figure 10.

The volume of the box is

Volume of box

To express as a function of the variable x alone, we need to eliminate L. We

know that , so . Using this expression for L, we

can now express the function as a function of x alone.

Volume of box

Replace L by

Distributive property

(b) The graph is shown in Figure 11. From the graph we see that the maximum

value occurs when x is 18.

Let’s find the length of the box.

From part (a)

Replace x by 18

Calculate

So the dimensions of the box with maximum volume are

inches.

■ NOW TRY EXERCISE 47 ■

18 * 18 * 36

= 36

= 108 - 4

L = 108 - 4x

V = 108x

- 4x

108 - 4x V = x

1108 - 4x 2

V = x

L = 108 - 4x4x + L = 108

V = x

figure 11 Graph of

V = 108x

- 4x

12,000

example

Solving a Polynomial Equation Graphically

If the box in Example 6 is to have a volume of 8000 cubic inches, what are the di-

mensions of the box?

Solution

From Example 6 we know that the volume of the box is modeled by the function

We graph the function and the line in Figure 12. From

the graph we see that the volume is 8000 when x is approximately 11.28 or 23.32.

To find the corresponding lengths L, we use the equation , which we

found in Example 6.

For we have .

So there are two possible boxes of the given shape having volume 8000 cubic inches.

Their dimensions are approximately

where all distances are measured in inches.

■ NOW TRY EXERCISE 49 ■

11.28 * 11.28 * 62.88or23.32 * 23.32 * 14.72

L L 108 - 4123.322= 14.72x L 23.32

L L 108 - 4111.282= 62.88x L 11.28

L = 108 - 4x

V = 8000V = 108x

- 4x

V = 108x

- 4x

12,000

11.28 23.32

figure 12

512 CHAPTER 6

■

Power, Polynomial, and Rational Functions

Fundamentals

1. (a) A polynomial function is a sum of _______ functions.

(b) The polynomial is the sum of the power functions

______, ______, and ______. The leading term of P is ______,

and the degree of P is

_______.

2. (a) To find the zeros of a polynomial function, we first

_______ the polynomial and

then apply the Zero-Product Property. The zeros of a polynomial are the

____-

intercepts of the graph of the polynomial.

(b) The polynomial function is in factored form.

The zeros of f are

____, ____, ____. The x-intercepts of the graph of f are ____,

____, ____.

3. Between consecutive x-intercepts the values of a polynomial must always be

_____________ or always be _____________. To decide which sign is the

appropriate one, we use

_______ points.

4. Every nonconstant polynomial has one of the following end behaviors:

(i)

(ii)

(iii)

(iv)

For each polynomial, choose the appropriate description of its end behavior from the

list above.

(a) ; end behavior:

__________________________.

(b) ; end behavior:

__________________________.

(d) ; end behavior:

__________________________.y =-5x

- 3x

+ 9x + 12

y = 5x

+ 3x

- 9x - 12

y =-x

+ 8x

- 2x + 15

y = x

- 8x

+ 2x - 15

y S

- q as x Sq and y S - q as x S - q

y S

- q as x Sq and y Sq as x S - q

y Sq as x Sq and y S

- q as x S - q

y Sq as x Sq and y Sq as x S

- q

f 1x 2= 1x - 221x + 121x - 32

y =y =y =

P1x 2= 3x

- 2x

+ 4

Check your knowledge of factoring polynomial expressions by doing the follow-

ing problems. You can review these topics in

Algebra Toolkit B.2

on page T33.

1. Factor out the common factor.

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

2. Factor the quadratic.

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

3. Factor the expression completely.

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

4. Factor the expression by grouping terms.

(a) (b)

- 14z

+ 5z

- 10x

+ 3x

+ x + 3

- 12r

+ 36x

+ 3x

- 28x

- 163t

- 12t

- n - 2t

+ 3t - 2825m

+ 10m + 1x

- 81

- 6y

+ 4yt

+ 4t

- 3r6x

+ 3

CONCEPTS

6.3 Exercises

SECTION 6.3

■

Polynomial Functions: Combining Power Functions 513

Think About It

5. Can the polynomial whose graph is shown have a degree that is even?

6. A function f is odd if or even if .

(a) Explain why a power function that contains only odd powers of x is an odd function.

(b) Explain why a power function that contains only even powers of x is an even

function.

neither an odd nor an even function

7–8

■ Graph the given polynomial function by expressing it as a sum of power functions,

and then using graphical addition.

7. 8.

9. Let .

(a) Show that .

(b) Find the x-intercepts of the graph of f.

(d) Sketch a graph of f.

10. Let .

(a) Show that .

(b) Find the x-intercepts of the graph of f.

(d) Sketch a graph of f.

11–18

■ Sketch a graph of the polynomial function. Make sure your graph shows all

intercepts and exhibits the proper end behavior.

11. 12.

13. 14.

15. 16.

17. 18.

19–26

■ A polynomial function P is given.

(a) Express the function P in factored form.

(b) Sketch a graph of P.

19. 20.

21. 22.

23. 24.

25. 26. P1x 2= x

- 2x

+ 1P1x 2= x

- 3x

- 4

P1x 2= x

- 9x

P1x 2= x

- 3x

+ 2x

P1x 2=-2x

- x

+ xP1x 2=-x

+ x

+ 12x

P1x 2= x

+ 2x

- 8xP1x 2= x

- x

- 6x

P1x 2= 1x - 32

1x + 12

P1x 2= 1x - 12

1x - 32

P1x 2=

x1x - 52

P1x 2= 1x - 321x + 2213x - 22

P1x 2= 12x - 121x + 121x + 32P1x 2= x1x - 3 21x + 22

P1x 2= 1x - 121x + 121x - 22P1x 2= 1x - 121x + 22

f 1x 2= 1x + 321x - 221x +

f 1x 2= x

+ 3x

- 4x - 12

f 1x 2= 1x + 12

1x - 12

f 1x 2= x

+ x

- x - 1

f 1x 2= x

+ 2x - 1f 1x2=-x

+ 2x

+ 5

f 1- x2= f 1x2f 1- x2=-f 1x2

SKILLS