This document contains information about polynomial functions including: - Examples of polynomial and non-polynomial functions along with their degrees - Properties of the graphs of polynomial functions such as intercepts, end behavior, and turning points - Finding a polynomial function given its zeros and verifying the result with a graphing utility - Describing zeros and their multiplicities for a polynomial function - Sketching graphs of polynomial functions based on their properties

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Section 4.1
Polynomial Functions and
Models

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( ) 3 8
(a) 3 4f x x x x= − +
( )(c) 5h x =
( )
2
3
(b)
1
x
g x
x
+
=
−
( )(d) ( 3)( 2)F x x x= − +
( ) 1
(e) 3 4G x x x−
= − ( ) 3 21 2 1
(f)
2 3 4
H x x x x= − +
(a) is a polynomial of degree 8.f (b) is not a polynomial function.
It is the ratio of two distinct polynomials.
g
( ) 0
(c) is a polynomial function of degree 0.
It can be written 5 5.
h
h x x= = 2
(d) is a polynomial function of degree 2.
It can be written ( ) 6.
F
F x x x= − −
(e) is not a polynomial function.
The second term does not have a
nonnegative integer exponent.
G
(f) is a polynomial of degree 3.H

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Summary of the Properties of the Graphs
of Polynomial Functions

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Find a polynomial of degree 3 whose zeros are -4, -2, and 3.
Use a graphing utility to verify your result.
( ) ( ) ( ) ( )4 2 3f x a x x x= + + −
( ) ( ) ( ) ( )4 2 3f x x x x= + + −
( ) ( ) ( ) ( )4 2 3f x x x x= − + + −
( ) ( ) ( ) ( )2 4 2 3f x x x x= + + −
( )3 2
3 10 24a x x x= + − −

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( ) ( ) ( ) ( )
3 4
2 2 1 3f x x x x= − − + −
For the polynomial, list all zeros and their multiplicities.
2 is a zero of multiplicity 1 because the exponent on the factor x – 2 is 1.
–1 is a zero of multiplicity 3 because the exponent on the factor x + 1 is 3.
3 is a zero of multiplicity 4 because the exponent on the factor x – 3 is 4.

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( ) ( )
2
3f x x x= −
( ) ( )
2 2
(a) -intercepts: 0 3 0 or 3 0x x x x x= − = − =
0 or 3x x= =
( ) ( )
2
-intercept: 0 0 0 3 0y f = − = 0y =

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( ) ( )
2
3f x x x= −
( ) ( )0,0 , 3,0
( ),0−∞ ( )0,3 ( )3,∞
1−
( )1 16f − = −
Below -axisx
( )1, 16− −
1
( )1 4f =
Above -axisx
( )1,4
4
( )4 4f =
Above -axisx
( )4,4

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( ),0−∞ ( )0,3 ( )3,∞
1−
( )1 16f − = −
Below -axisx
( )1, 16− −
1
( )1 4f =
Above -axisx
( )1,4
4
( )4 4f =
Above -axisx
( )4,4
( ) ( )
2
3f x x x= −
−3 −2 −1 1 2 3 4 5 6
−4
−3
−2
−1
1
2
3
4
x
y

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y = 4(x - 2)

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y = 4(x - 2)

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( ) ( )
2
3f x x x= −

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Figure 16 (a)

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Figure 16 (b)

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Figure 16 (c)

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Figure 16 (d)

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( )0 6 so the intercept is 6.f y= − −
The degree is 4 so the graph can turn at most 3 times.
4
For large values of , end behavior is like (both ends approach ).x x ∞

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1
The zero has multiplicity 1
2
so the graph crosses there.
−
The zero 3 has multiplicity 2
so the graph touches there.

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The polynomial is degree 3 so the
graph can turn at most 2 times.

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