copyright © 2013, 2009, 2005 pearson education, inc. section 5.1 polynomial functions

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

Section 5.1

Polynomial Functions


Objectives

• Monomials and Polynomials

• Addition and Subtraction of Polynomials

• Polynomial Functions

• Evaluating Polynomials

• Operations on Functions

• Applications and Models


Monomials and Polynomials

A term is a number, a variable, or a product of numbers and variables raised to powers.

Examples of terms:

If the variables in a term have only nonnegative integer exponents, the term is called a monomial.

Examples of monomials:3 2 9 88, 7 , , 8 , y x x y xy

5 1/2 2 1 812, , , 8 , 8z x x y x y


Example

Determine whether the expression is a monomial. a. b. c. d.

Solutiona. b.

c. d.

4 79x y 3ab 48w x 3

y

4 79x y 3ab

48w x3

y

monomialnot a monomialnegative exponent

not a monomialsum (+) of two monomials

not a monomialnegative exponent y-1 since 3/y = 3y-1

Division by a variable


Monomials

The degree of a monomial equals the sum of the exponents of the variables. A constant term has degree 0, unless the term is 0 (which as an undefined degree). The numeric constant in a monomial is called its coefficient. The table shows the degree and coefficient of several monomials.


Polynomials

A polynomial is either a monomial or a sum of monomials.Polynomials containing one variable are called polynomials of one variable. The leading coefficient of a polynomial of one variable is the coefficient of the monomial with highest degree.The degree of a polynomial equals the degree of the monomial with the highest degree.


Addition and Subtraction

We can add like terms.If two terms contain the same variables raised to the same power, we call them like terms.


Example

Simplify each expression by combining like terms.a. b.

Solutiona.

b.

2 2 47 3x x x 2 2 2 28 3 2ab a ab a

2 2 47 3x x x 2 4( )7 3 x x

2 44x x

2 2 2 28 3 2ab a ab a 2 2 2 28 13 2ab a ab a

2 2( )1 3( 28 )ab a

2 27ab a


Example

Simplify the expression.

Solution

2 23 4 8 4 5 3x x x x

2 28 3443 5x x x x

2 2 4 8 34 53x x x x

2 23 4 8 4 5 3x x x x

2 4( ) (3 4 )3) (85x x

2 57x x


Example

Find the sum.

SolutionPolynomials can be added vertically by placing like terms in the same columns and then adding each column.

2 2 2 27 3 7 2 2x xy y xy y x

2

2

2

2

7 3 7

2 2

yxy

yx y

x

x

2 25 2 5xyx y


Subtracting Polynomials

To subtract two polynomials we add the first polynomial to the opposite of the second polynomial. To find the opposite of a polynomial, we negate each term.


Example

Simplify.SolutionThe opposite of

3 2 3 25 3 6 5 4 8w w w w

3 2 3 25 4 8 is 5 4 8w w w w

3 2 3 25 3 6 5 4 8w w w w

3 2(5 5) (3 4) ( 6 8)w w

3 20 7 2w w 27 2w


Polynomial Functions

The following expressions are examples of polynomials of one variable.

As a result, we say that the following are symbolic representations of polynomial functions of one variable.

2 3, , and 51 515 3x xx x

32( ) 3, , and ( 5 1 ( )) 1 5 5f g x x xx xx hx


Example

Determine whether f(x) represents a polynomial function. If possible, identify the type of polynomial function and its degree. a.

b.

c.

3( ) 6 2 7f x x x

3.5( ) 4f x x

4( )

5f x

x

cubic polynomial, of degree 3

not a polynomial function because the variable is negative

not a polynomial


Example

A graph of is shown. Evaluate f(1) graphically and check your result symbolically.

3( ) 5f x x x

To calculate f(–1) graphically find –1 on the x-axis and move down until the graph of f is reached. Then move horizontally to the y-axis.

f(1) = –4 3( 1) 5( ( )1 1)f

5 1

4


Example

Evaluate f(x) at the given value of x.

Solution

3 2( ) 4 3 7, 2f x x x x

3 2( ) 4 3 7, 2f x x x x

3 2( 2) 4( 2) 3( 2) 7f

( 2) 4( 8) 3(4) 7f

( 2) 32 12 7f

( 2) 27f


Example

Let f(x) = 3x2 + 1 and g(x) = 6 – x2. Find each sum or difference. Solutiona.

b.

2 ( ) 3(

4

1 1) 1

f 2( ) 61 ( )

5

1g

2( ) 3( ) 1

3(4) 1

1

2

3

2

f 2( ) 6 (

6 4

2

2 )2g

( )( 2) ( 2) ( 2)

13 2

11

f g f g

( )(1) (1) (1)

4 5

9

f g f g

a. ( )(1) b. ( )( 2)f g f g


Example

Let model an athlete’s heart rate (or pulse) in beats per minute (bpm) t minutes after strenuous exercise has stopped, where 0 ≤ t ≤ 8. a. What is the initial heart rate when the athlete stops exercising?

2( ) 1.875 30 200P t t t

2( ) 1.875 30 200 P t t t

When the athlete stops exercising, the heart rate is 200 beats per minute.

2(0) 1.875(0) 30(0) 200

(0) 200

P

P


Example (cont)

Let model an athlete’s heart rate (or pulse) in beats per minute (bpm) t minutes after strenuous exercise has stopped, where 0 ≤ t ≤ 8. b. What is the heart rate after 8 minutes?

2( ) 1.875 30 200P t t t

2( ) 1.875 30 200 P t t t2(8) 1.875(8) 30(8) 200

(0) 80 beats per minute.

P

P


Example (cont)

Let model an athlete’s heart rate (or pulse) in beats per minute (bpm) t minutes after strenuous exercise has stopped, where 0 ≤ t ≤ 8. c. A graph of P is shown. Interpret this graph.

The heart rate does not drop at a constant rate; rather, it drops rapidly at first and then gradually begins to level off.

2( ) 1.875 30 200P t t t

copyright © 2013, 2009, 2005 pearson education, inc. section 5.1 polynomial functions

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