Polynomial Functions End Behavior Section 2-2 2 Objectives I can determine if an equation is a...

Polynomial FunctionsEnd Behavior

Section 2-2

2

Objectives

• I can determine if an equation is a polynomial in one variable

• I can find the degree of a polynomial• I can use the Leading Coefficient Test for

end behavior in Limit Notation

3

A polynomial function is a function of the form1

1 1 0( ) , 0n nn n nf x a x a x a x a a

where n is a nonnegative integer and each ai (i = 0, , n)

is a real number. The polynomial function has a leading coefficient an and degree n.

Examples: Find the leading coefficient and degree of each polynomial function.

Polynomial Function Leading Coefficient Degree5 3( ) 2 3 5 1f x x x x

3 2( ) 6 7f x x x x ( ) 14f x

– 2 5

1 3

14 0

4

Complex Numbers

Real Numbers Imaginary Numbers

Rationals Irrational

5

Polynomial Functions/Equations:

A polynomial function in one variable may look like this.

5 4 3 25 2 4 3f x x x x x A.The coefficients are complex numbers (real or imaginary).

B. Exponents must be a non-negative integer (zero or positive).

C.The leading coefficient (the coefficient of the variable with greatest degree) may not be zero.

6

Polynomials Not a polynomial

23 1f x x x

53f x x

23

15

xf x

1/ 23 3f x x x the exp. is not an integer

134 3 4f x x xx

x

the exp. is not non-negative

2

5

4

xf x

x

denominator has a variable factor

7

EX: Determine if each expression is a polynomial in one variable

4 22 4f x x x x YES

4 2 5 35 6f s s s s s YES

No 2 33 9f y y

y

8

Practice 1: Given the following equations determine the following:

1. Determine if the equation a polynomial. Why or why not?

2. If the equation is a polynomial what is the degree of each term, of the polynomial.

23 1f x x x 34f x x

x

3f x x 23

15

xf x

A. B.

C. D.

Yes, notice powers on the x are positive integers and coefficientsare real numbers.

No, notice power on the x is the fraction 1/2 1

2x x

No, notice power on x is -1

133x

x

Yes, notice powers on the x are positive integers and coefficientsare real numbers.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9

Group ExplorationDirections:Divide into groups of 2Open your text to page 141. Read the Exploration exercise

10 minutes!!

Answers on next slides

10

Group ExplorationOpen the text to page 141.

Read the Exploration exercise instructions.

Use the Leading coefficient test on page 141

a.

b.

c.

d.

e.

f.

g.

3 2( ) 2 1f x x x x The leading coefficient is + 1 The degree of the function is 3, that is, f(x) is a cubic.

( ) as x -

f(x) as x

f x





b.

c.

d.

e.

f.

g.

5 2( ) 2 2 5 1f x x x x

( ) as x -

f(x) as x

f x

The leading coefficient is + 2The degree is 5 and odd.

12




c.

d.

e.

f.

g.

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

( ) as x

f(x) as x

f x

The leading coefficient is - 2The degree is 5 and odd.

13




d.

e.

f.

g.

3( ) 5 2f x x x

( ) as x

f(x) as x

f x

The leading coefficient is - 1The degree is 3 and odd.

14




e.

f.

g.

2( ) 2 3 4f x x x

The leading coefficient is + 2The degree is 2 and even.

f(x) + as x -

f(x) + as x +

15




f.

g.

4 2( ) 3 2 1f x x x x The leading coefficient is + 1The degree is 4 and even.

f(x) + as x -

f(x) + as x +





g. 2( ) 3 2f x x x

The leading coefficient is + 1The degree is 2 and even.

f(x) + as x -

f(x) + as x +


Leading Coefficient TestAs x grows positively or negatively without bound, the value f (x) of the polynomial function

f (x) = anxn + an – 1x

n – 1 + … + a1x + a0 (an 0)

grows positively or negatively without bound depending upon the sign of the leading coefficient an and whether the degree n

is odd or even.

x

y

x

y

n odd n even

an positive

an negative

18

Example: Describe the right-hand and left-hand behavior for the graph of f(x) = –2x3 + 5x2 – x + 1.

As , and as , xx )(xf )(xf

Negative-2Leading Coefficient

Odd3Degree

x

y

f (x) = –2x3 + 5x2 – x + 1

19

Closure:

20

A real number a is a zero of a function y = f (x)if and only if f (a) = 0.

Real Zeros of Polynomial FunctionsIf y = f (x) is a polynomial function and a is a real number then the following statements are equivalent.

1. (a, 0) is a zero of f.2. x = a is a solution of the polynomial equation f (x) = 0.3. (x – a) is a factor of the polynomial f (x).

4. (a, 0) is an x-intercept of the graph of y = f (x).

21

Solution or Root Zero or X-intercept Factor

4x (4,0) ( 4)x

2x 2

3x

( 2,0)2

( ,0)3

( 2)x

(3 2)x

22

Homework

• WS 3-3

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