Section 5.1

Polynomial Functions and Models

Polynomial Functions

Three of the families of functions studied thus far:

constant, linear, and quadratic, belong to a much

larger group of functions called polynomials.

We begin our formal study of general polynomials

with a definition and some examples.

Polynomial Functions

A polynomial function is a function of the form

f (x) an xn + an1 xn1 + … + a2 x2 + a1 x + a0

where a0, a1, . . . , an are real numbers and n 1 is

a natural number.

The domain of a polynomial function is ( , ).

Polynomial Functions

Suppose f is the polynomial function

f (x) an xn + an1 xn1 + … + a2 x2 + a1 x + a0

where an 0. We say that, The natural number n is the degree of the polynomial f. The term anxn is the leading term of the polynomial f.

The real number an is the leading coefficient of the polynomial f.

The real number a0 is the constant term of the polynomial f.

If f (x) a0, and a0 0, we say f has degree 0.

If f (x) 0, we say f has no degree.

Determine which of the following functions are

polynomials. For those that are, state the degree.

3 8(a) 3 4f x x x x

(c) 5h x

2 3

(b) 1

xg x

x

(d) ( 3)( 2)F 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

Identifying Polynomial Functions

1(e) 3 4G x x x 3 21 2 1(f)

2 3 4H x 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

Determine which of the following functions are

polynomials. For those that are, state the degree.

Identifying Polynomial Functions

Polynomial Functions: Example

A box with no top is to be built from a 10 inch by

12 inch piece of cardboard by cutting out congruent

squares from each corner of the cardboard and then

folding the resulting tabs.

Let x denote the length of the side of the square

which is removed from each corner.

A diagram representing the situation is,

Polynomial Functions: Example

1. Find the volume V of the box as a function of x. Include an appropriate applied domain.

2. Use a graphing calculator to graph y V (x) on the domain you found in part 1 and approximate the dimensions of the box with maximum volume to two decimal places. What is the maximum volume?

Polynomial Functions: Example

Summary of the Properties of the Graphs of Polynomial Functions

Graphs of Polynomial Functions

Power Functions

A power function of degree n is a function of the

form

f (x) axn

where a 0 is a real number and n 1 is an

integer.

Power Functions: a 1, n even

Power Functions: a 1, n even

Power Functions: a 1, n even

Power Functions: a 1, n odd

Power Functions: a 1, n odd

Power Functions: a 1, n odd

Identifying the Real Zeros of a Polynomial Function and

Their Multiplicity

Graphs of Polynomial Functions

Definition: Real Zero

Find a polynomial of degree 3 whose zeros are

4, 2, and 3.

4 2 3f x a x x x 3 23 10 24a x x x

Finding a Polynomial Function from Its Zeros

The value of the leading coefficient a is, at this point, arbitrary. The next slide shows the graph of three polynomial functions for different values of a.

4 2 3f x x x x

4 2 3f x x x x

2 4 2 3f x x x x

Finding a Polynomial Function from Its Zeros

3 42 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.

Definition: Multiplicity

23f 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

Graphing a Polynomial UsingIts x-Intercepts

23f 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

23f x x x

x

y

,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

Behavior Near a Zero

Example

y = 4(x - 2)

Example

y = 4(x - 2)

Turning Points: Theorem

End Behavior

End Behavior: Example

End Behavior: Example

0 6 so the intercept is 6.f y The degree is 4 so the graph can turn at most 3 times.

4For large values of , end behavior is like (both ends approach )x x

Summary

Analyze the Graph of a Polynomial Function

1The zero has multiplicity 1

2so the graph crosses there.

The zero 3 has multiplicity 2

so the graph touches there.

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

Summary: Analyzing the Graph of a Polynomial Function

The domain and the range of f are the set of all real numbers.

Decreasing: 2.28,0.63

Increasing: , 2.28 and 0.63,