2.32.32.32.3

Real Zeros of Polynomial Real Zeros of Polynomial FunctionsFunctions

Quick Review

3 2

5 3 2

2

3

3 2

2

Rewrite the expression as a polynomial in standard form.

2 31.

2 82.

2Factor the polynomial into linear factors.

3. 16

4. 4 4

5. 6 24

x x x

xx x x

x

x x

x x x

x

Quick Review Solutions

3 2

5 3 2

2

3

3 2

2

3

Rewrite the expression as a polynomial in standard form.

2 31.

2 82.

2Factor the polynomial into linear factors.

3. 16

4. 4

2 3 1

14

2

4 4

4 2 1 2

x x

x x

x

x x x

xx x x

x

x x

x x

x x

x x xx

25. 6 24 2 2 6 x xx

What you’ll learn about

• Long Division and the Division Algorithm• Remainder and Factor Theorems• Synthetic Division• Rational Zeros Theorem• Upper and Lower Bounds

… and whyThese topics help identify and locate the real

zeros of polynomial functions.

Example Using Polynomial Long

Division4 3

2

Use long division to find the quotient and remainder when 2 3

is divided by 1.

x x

x x

Example Using Polynomial Long

Division

4 3

2

Use long division to find the quotient and remainder when 2 3

is divided by 1.

x x

x x

2

2 4 3 2

4 3 2

3 2

3 2

2

2

4 3 2 2

2

2 11 2 0 0 3

2 2 2

2 0 3

+ 3

1

2 2

2 22 3 1 2 1

1

x xx x x x x x

x x x

x x x

x x x

x x

x x

x

xx x x x x x

x x

Remainder Theorem If polynomial ( ) is divided by , then the remainder is ( ).f x x k r f k

Example Using the Remainder Theorem

2Find the remainder when ( ) 2 12 is divided by 3.f x x x x

Example Using the Remainder Theorem

2Find the remainder when ( ) 2 12 is divided by 3.f x x x x

2

( 3) 2 3 3 12 =33r f

Example Using Synthetic Division

3 2Divide 3 2 5 by 1 using synthetic division.x x x x

Example Using Synthetic Division

3 2Divide 3 2 5 by 1 using synthetic division.x x x x

1 3 2 1 5

3

1 3 2 1 5

3 1 2

3 1 2 3

3 2

23 2 5 33 2

1 1

x x xx x

x x

Example Finding the Real Zeros of a Polynomial

Function4 3 2Find all of the real zeros of ( ) 2 7 8 14 8.f x x x x x

Example Finding the Real Zeros of a Polynomial

Function

4 3 2Find all of the real zeros of ( ) 2 7 8 14 8.f x x x x x

:

Factors of 8 1, 2, 4, 8 1: 1, 2, 4, 8,

Factors of 2 1, 2 2

Compare the -intercepts of the graph and the list of possibilities,

and decide that 4 and -1/2 are potential rational

Potential Rational Zeros

x

zeros.

Example Finding the Real Zeros of a Polynomial

Function4 3 2Find all of the real zeros of ( ) 2 7 8 14 8.f x x x x x

Example Finding the Real Zeros of a Polynomial

Function

4 3 2 3 2

4 2 7 8 14 8

8 4 16 8

2 1 4 2 0

This tells us that 2 7 8 14 8 ( 4)(2 4 2).

1/ 2

x x x x x x x x

4 3 2 2

2 1 4 2

1 0 2

2 0 4 0

1This tells us that 2 7 8 14 8 2( 4) 2 .

2

1The real zeros are 4,

x x x x x x x

, 2.2

4 3 2Find all of the real zeros of ( ) 2 7 8 14 8.f x x x x x