Chapter 4

Polynomial (Poly) & Rational Functions

Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc.

Sec 4.1

Polynomial Functions and Modeling

A polynomial function P is given by:

where the coefficients an, an - 1, …, a1, a0 are real numbers and the exponents are Whole numbers (0,1,2,….)

The graph of a polynomial function is continuous and smooth.

The domain of a polynomial function is the set of all real numbers (ARN).

Polynomial Function

1 21 2 1 0( ) ... ,n n n

n n nP x a x a x a x a x a

Examples of Polynomial Functions

Examples of Nonpolynomial Functions

Quadratic Function

Cubic Function

The Leading-Term Test

Match each of the following functions with one of the graphs AD

a.

b.

c.

d.

4 3( ) 3 2 3f x x x

3 2( ) 5 4 2f x x x x

5 14( ) 1f x x x

6 5 3( ) 4f x x x x

Graphs

4 3( ) 3 2 3f x x x

3 2( ) 5 4 2f x x x x 5 14( ) 1f x x x

6 5 3( ) 4f x x x x

Finding Zeros of Factored Polynomial Functions

If c is a real zero of a function that is, f (c) = 0, then:

(c, 0) is a “zero” (x-intercept) of the function, and (x-c) is a “factor” of the function.

Example

Find the zeros of

To solve the equation f(x) = 0, we use the principle of zero products, solving x 1 = 0 and x + 2 = 0.

The zeros of f(x) are 1 and 2.

3

( ) 5( 1)( 1)( 1)( 2)

5( 1) ( 2).

f x x x x x

x x

Even and Odd “Multiplicity”

If (x c)k, where k 1, is a factor of a polynomial function

P(x) and:

k is odd, then the graph crosses the x-axis at (c, 0);

k is even, then the graph is tangent (touches and reverses, but does not cross) to the x-axis at (c, 0).

Example

Find the zeros of f (x) = x3 – 2x2 – 9x + 18.Solution We factor:f (x) = x3 – 2x2 – 9x + 18 = x2(x – 2) – 9(x – 2).

2( 2)( 9)

( 2)( 3)( 3)

x x

x x x

By the Principle of Zero Products, the solutions of the equation f(x) = 0, are 2, –3, and 3 and the Zeros are (2,0), (-3,0) and (3,0). Since these are all factors of multiplicity 1, the curve crosses the Horiz axis at each point.

Example

Find the zeros of f (x) = x4 + 8x2 – 33.

We factor as follows:f (x) = x4 + 8x2 – 33 = (x2 + 11)(x2 – 3).

Solve the equation f(x) = 0 to determine the zeros. 2 2

2 2

2 2

2

( 11)( 3) 0

11 0 or 3 0

11 or 3

11 or 3

11

x x

x x

x x

x x

x i

Example

Find the zeros of f (x) = 0.2x3 – 1.5x2 – 0.3x + 2.Approximate the zeros to three decimal places.

Solution Use a graphing calculator to create a graph. Look for points where the graph crosses the x-axis. We use the ZERO feature to find them.

The zeros are approximately –1.164, 1,142, and 7.523.

–1010

Example

The polynomial function

can be used to estimate the number of milligrams (M in mg) of the pain relief medication “ibuprofen” in the bloodstream t (hrs) after 400 mg of the medication has been taken. Find the number of mg of the drug in the blood at t = 0, 0.5, 1, 1.5, and so on, up to 6 hr. (use one decimal accuracy)

4 3 2( ) 0.5 3.45 96.65 347.7 M t t t t t

Example - cont

Using a calculator, we compute the function values.(0) 0

(0.5) 150.2

(1) 255

(1.5) 318.3

(2) 344.4

(2.5) 338.6

(3) 306.9

M

M

M

M

M

M

M

(3.5) 255.9

(4) 193.2

(4.5) 126.9

(5) 66

(5.5) 20.2

(6) 0

M

M

M

M

M

M