1

Warm-up• Determine if the following are polynomial functions

in one variable. If yes, find the LC and degree

xxxxxf

2

124316)( 243

225 32412)( xxxxg

Given the following polynomial function, find the following

information to help graph it! Degree? LC? End Behavior? Y-int, and

Factors with Multiplicity? Graph it!

234 6)( xxxxf

2

Warm-up

xxxxxf

2

124316)( 243

225 32412)( xxxxg

234 6)( xxxxf

NO

Yes, LC: -4;

Degree: 5

Degree: 4;

LC 1

Y-Int: (0, 0)

Factors: x2(x-3)(x+2)

x = 0 (m=2, even)

x = 3 (m = 1; odd)

x = -2 (m = 1; odd)

End Behavior

Left: Up

Right: Up

Using Synthetic Division to find Zeros

Section 2-3

4

Objectives

• I can use synthetic division to find factors of a polynomial

• I can use synthetic division to find zeros of a polynomial

5

Dividing Numbers

4164

When you divide a number by another number and there is NO REMAINDER:

Then the DIVISOR is a factor!!

Also the QUOTIENT becomes another factor!!!

Dividend

Divisor

Quotient

6

Find: (6x3- 19x2 + x + 6) (x-3)

• 6x3 – 19x2 + 1x + 6

6 -19 1 6

3

6

18

-1

-3

-2

-6

0

6x2 – 1x – 2 (No remainder)

That means (x – 3) is a factor and (6x2 – x – 2) is also a factor

That means (3, 0) is a zero.

7

Find: (4x4- 5x2 + 2x + 4) (x+1)

• 4x4 + 0x3 – 5x2 + 2x + 4

4 0 -5 2 4

-1

4

-4

-4

4

-1

1

3

-3

1

That means (x + 1) is NOT a factor

1

13144 23

xxxx

8

Finding Additional Factors or Zeros

• Sometimes you will know one factor or zero, but need to find the remaining factors or zeros

• Then using synthetic division we would divide by the known factor or zero and the quotient will be a new factor.

9

Given: x3 - x2 - 5x - 3; (x + 1)

1x3 – 1x2 – 5x – 3

-1

1 -1 -5 -3

1

-1

-2

2

-3

3

0

1x2 – 2x - 3

1x2 – 2x - 3

( )( )

(x – 3)(x + 1)

(3, 0) (-1, 0)

10

Given: x3 + 5x2 - 12x - 36; (3, 0)

1x3 + 5x2 – 12x – 36

3

1 5 -12 -36

1

3

8

24

12

36

0

1x2 + 8x - 12

1x2 + 8x - 12

( )( )

(x + 6)(x + 2)

(-6, 0) (-2, 0)

11

Factor Theorem: A polynomial f(x) has a factor (x – k) if and

only if f(k) = 0.

Example: Show that (x + 2) and (x – 1) are factors of

f(x) = 2x 3 + x2 – 5x + 2.

6

2 1 – 5 2– 2

2

– 4

– 3 1

– 2

0

The remainders of 0 indicate that (x + 2) and (x – 1) are factors.

– 1

2 – 3 11

2

2

– 1 0

The complete factorization of f is (x + 2)(x – 1)(2x – 1).

Read the Question

• Find the remaining factors

• Find all the factors

• Find the remaining zeros

• Find all the zeros

12

13

Homework

• WS 4-2