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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