Section 3.3 – Polynomials and Synthetic Division

3 2

2

3x x 4x 1

x 1

2 3 2x 1 3x x 4x 1

3

2

3x

x3x

3x

33x 3x2x x 1

2 3 2x 1 3x x 4x 1 3x

33x 3x2x x 1 2

21

x

x

1

2x 1

x

2

x3x 1

x 1

4 2

2

x 2x x 3

x x 1

2 4 230xx x 1 x 2x x 3

4

22x

xx

4x 3x3 2x x x

2x x

2x

2x

2 4 3 2x x 1 x 0x 2x x 3 4x 3x

3 2x x x

2x

2x

3

2x

x

x

x

3x 2x x

3 2x 2 x 0x x 2 3x

22x x 2

2x

22x

3x x 2

x 2

3 20x 2 x 2xx

23x

xx

3x22x x 2

2x

22x

22x

x2x

2x

22x 4x5x 25x

x5

5

5x 10

12

2 12x 2x 5

x 2

3 2x 4x 2x 5

x 3

3 2x 3 x 4x 2x 5

23x

xx

3 2x 3x2x 2x 5

2x

2

xx

x

x

2x 3x

x 5

1x

x

1

x 3 8

2 8x x 1

x 3

3 22x x 2x 3

x 1

3 2x 1 2x x 2x 3

322x

x2x

3 22x 2x23x 2x 3

22x

23x

x3x

3x

23x 3x

x 3

1x

x

1

x 1 4

2 42x 3x 1

x 1

3 2x 4x 2x 5

x 3

x

x

3

3

0

1 -4 2 -5 3

1

3

-1

-3

-1

-3

-8

2x x 1x

8

3

3 22x x 2x 3

x 1

x

x

1

1

0

2 -1 2 -3 1

2

2

1

1

3

3

0

22x x 3

3 24x 3x 8x 4

x 3

x

x

3

3

0

4 -3 -8 4 3

4

12

9

27

19

57

61

24x 9x 161

9x 3

3 22x 5x 28x 14

x 5

x

x

5

5

0

2 -5 -28 14 5

2

10

5

25

-3

-15

-1

22x 5xx

13

5

3 216x 32x 81x 162

x 2

x

x

2

2

0

16 -32 -81 162 2

16

32

0

0

-81

-162

0

216x 81

3 2x 2x x 1

x 3

x

x

3

3

0

1 -2 -1 1 3

1

3

1

3

2

6

7

2x x 2x

7

5

3x 5x 2

x 3

x

x

3

3

0

1 0 -5 2 3

1

3

3

9

4

12

14

2x 3x 4x

4

3

1

4 2x 17x 16

x 4

x

x

4

4

0

1 0 -17 0 16 4

1

4

4

16

-1

-4

-4

3 2x 4x x 4

23x 5

x

0 x 2

3

x

4 23x 17x 1x 60

4

0x

x

-16

0

3 26x 4x 3x 2

3x 2

3x 2 0

2

3x

6 -4 3 -2 2/3

6

4

0

0

3

2

0

22x 1

4 3 22x 5x 4x 5x 2

2x 1

2x 1 0

1x

2

2 5 4 5 2

2

-1

4

-2

2

-1

4

3 2x 2x x 2

-2

0

3

-1/2

2

3 24x x 4x 1

4x 1

4x 1 0

1

4x

4 -1 -4 1 1/4

4

1

0

0

-4

-1

0

2x 14

3 21Find f if f x 4x x 4x 1

4

1f

40

34x 13x 6

2x 1

2x 1 0

1x

2

4 0 -13 -6

4

-2

-2

1

-12

6

0

22x x 6

-1/2

2

31Find f if f x 4x 13x 6

2

f 01

2

3

x 3

x 5x 2

x 3 0

x 3

1 0 -5 2 3

1

3

3

9

4

12

14

2 14x 3x 4

x 3

3 2x 0x 5x 2

x 3

3Find f 3 if f x x 5x 2

f 3 14

4 2x 17x

x 4

16

x 4 0

x 4

1 0 -17 0 16 4

1

4

4

16

-1

-4

-4

3 2x 4x x 4

4 3 2x 0x 17x 0x 16

x 4

-16

0

4 2Find f 4 if f x x 17x 16

f 4 0

Synthetic Division Summary

1. Set denominator = 0 and solve (box number)2. Bring down first number3. Multiply by box number and add until finished4. Remainder goes over divisor

Notes of Caution

1. ALL terms must be represented (even if coefficient is 0)2. If box number is a fraction, must divide final answer by the denominator

To evaluate a function at a particular value, you may EITHER:A) Substitute the value and simplify ORB) Complete synthetic division…the remainder is your answer