This document discusses polynomial functions. It defines a polynomial function as a function of the form f(x) = anxn + an-1xn-1 + ... + a1x + a0, where an ≠ 0 and n is a whole number. The document provides examples of evaluating polynomial functions, using polynomials to model data, adding and subtracting polynomials, and graphing basic polynomial functions like identity, square, and cube functions. The objectives are to recognize and evaluate polynomial functions, use them to model data, add and subtract them, and graph basic polynomial functions.

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Sec 6.3 - 1

2. Copyright © 2010 Pearson Education, Inc. All rights reserved
Sec 6.3 - 2
Exponents, Polynomials, and
Polynomial Functions
Chapter 6

3. Copyright © 2010 Pearson Education, Inc. All rights reserved
Sec 6.3 - 3
6.3
Polynomial Functions

4. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 4
6.3 Polynomial Functions
Objectives
1. Recognize and evaluate polynomial functions.
2. Use a polynomial function to model data.
3. Add and subtract polynomial functions.
4. Graph basic polynomial functions.

5. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 5
6.3 Polynomial Functions
Definition of a Polynomial Function
Polynomial Function
A polynomial function of degree n is defined by
f (x) = an xn + an – 1 xn – 1 + · · · + a1 x + a0 ,
for real numbers an,an – 1, . . . , a1, and a0 , where an ≠ 0 and n is a whole
number.

6. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 6
6.3 Polynomial Functions
EXAMPLE 1 Evaluating Polynomial Functions
Let f(x) = 4x3 – 5x2 + 7. Find each value.
(a) f(2)
f(x) = 4x3 – 5x2 + 7
f(2) = 4 • 23 – 5 • 22 + 7
= 4 • 8 – 5 • 4 + 7
= 32 – 20 + 7
= 19

7. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 7
6.3 Polynomial Functions
EXAMPLE 1 Evaluating Polynomial Functions
Let f(x) = 4x3 – 5x2 + 7. Find each value.
(b) f(–3)
f(x) = 4x3 – 5x2 + 7
f(–3) = 4 • (–3)3 – 5 • (–3)2 + 7
= 4 • (–27) – 5 • 9 + 7
= –108 – 45 + 7
= –146

8. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 8
6.3 Polynomial Functions
Functions
While f is the most common letter used to represent functions, recall that
other letters such as g and h are also used. The capital letter P is often used
for polynomial functions.

9. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 9
6.3 Polynomial Functions
EXAMPLE 2 Using a Polynomial Model to Approximate
Data
The number of U.S. households estimated to see and pay at least one bill
on-line each month during the years 2000 through 2006 can be modeled by
the polynomial function defined by
P(x) = 0.808x2 + 2.625x + 0.502,
where x = 0 corresponds to the year 2000, x = 1 corresponds to 2001, and
so on, and P(x) is in millions. Use this function to approximate the number
of households expected to pay at least one bill on-line each month in 2006.
Since x = 6 corresponds to 2006, we must find P(6).
P(x) = 0.808x2 + 2.625x + 0.502
P(6) = 0.808(6)2 + 2.625(6) + 0.502
= 45.34
Thus, in 2006 about 45.34 million households are expected to pay at least
one bill on-line each month.
Let x = 6.
Evaluate.

10. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 10
6.3 Polynomial Functions
Adding and Subtracting Functions
Adding and Subtracting Functions
If f(x) and g(x) define functions, then
(f + g) (x) = f (x) + g(x) Sum function
and (f – g) (x) = f (x) – g(x). Difference function
In each case, the domain of the new function is the intersection of the
domains of f(x) and g(x).

11. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 11
6.3 Polynomial Functions
EXAMPLE 3 Adding and Subtracting Functions
For the polynomial functions defined by
f(x) = 2x2 – 3x + 4 and g(x) = x2 + 9x – 5,
find (a) the sum and (b) the difference.
(a) (f + g) (x) = f (x) + g(x) Use the definition.
= (2x2 – 3x + 4) + (x2 + 9x – 5) Substitute.
= 3x2 + 6x – 1 Add the polynomials.
(b) (f – g) (x) = f (x) – g(x) Use the definition.
= (2x2 – 3x + 4) – (x2 + 9x – 5) Substitute.
= (2x2 – 3x + 4) + (–x2 – 9x + 5) Change subtraction
to addition.
= x2 – 12x + 9 Add.

12. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 12
6.3 Polynomial Functions
EXAMPLE 4 Adding and Subtracting Functions
For the polynomial functions defined by
f(x) = 4x2 – x and g(x) = 3x,
find each of the following.
(a) (f + g) (5)
(f + g) (5) = f (5) + g(5) Use the definition.
= [4(5)2 – 5] + 3(5) Substitute.
= 110

13. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 13
6.3 Polynomial Functions
EXAMPLE 4 Adding and Subtracting Functions
For the polynomial functions defined by
f(x) = 4x2 – x and g(x) = 3x,
find each of the following.
(a) (f + g) (5)
(f + g) (x) = f (x) + g(x) Use the definition.
= (4x2 – x) + 3x Substitute.
= 4x2 + 2x
Alternatively, we could first find (f + g) (x).
Then,
(f + g) (5) = 4(5)2 + 2(5) = 110. The result is the same.

14. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 14
6.3 Polynomial Functions
EXAMPLE 4 Adding and Subtracting Functions
For the polynomial functions defined by
f(x) = 4x2 – x and g(x) = 3x,
find each of the following.
(b) (f – g) (x) and (f – g) (3)
(f – g) (x) = f (x) – g(x) Use the definition.
= (4x2 – x) – 3x Substitute.
= 4x2 – 4x Combine like terms.
Then,
(f – g) (3) = 4(3)2 – 4(3) = 24. Substitute.
Confirm that f (3) – g(3) gives the same result.

15. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 15
6.3 Polynomial Functions
Basic Polynomial Functions
The simplest polynomial function is the identity function, defined by f(x) = x.
x
y
x f(x) = x
–2
–1
0
1
2
–2
–1
0
1
2

16. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 16
6.3 Polynomial Functions
Basic Polynomial Functions
The squaring function, is defined by f(x) = x2.
x
y
x f(x) = x2
–2
–1
0
1
2
4
1
0
1
4

17. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 17
6.3 Polynomial Functions
Basic Polynomial Functions
The cubing function, is defined by f(x) = x3.
x
y
x f(x) = x3
–2
–1
0
1
2
–8
–1
0
1
8

18. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 18
x f(x) = –2x
6.3 Polynomial Functions
EXAMPLE 5 Graphing Variations of the Identity Function
Graph the function by creating a table of ordered pairs. Give the domain and
the range of the function by observing the graph.
(a) f(x) = –2x.
x
y
–2
–1
0
1
2
4
2
0
–2
–4
Domain
Range

19. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 6.3 - 19
x f(x) = x2 – 2
6.3 Polynomial Functions
EXAMPLE 5 Graphing Variations of the Identity Function
Graph the function by creating a table of ordered pairs. Give the domain and
the range of the function by observing the graph.
(b) f(x) = x2 – 2.
x
y
–2
–1
0
1
2
2
–1
–2
–1
2
Domain
Range