1.
Objective:
Be able to factor a polynomial using synthetic division.
Critical Vocabulary:
Synthetic Division
Example 1: Factor: x3 + 2x2 - 5x - 6
x2(x + 2) -1(5x + 6)
p: +/-1, +/-2, +/-3, +/-4, +/-6, +/- 12
q: +/-1
p/q: +/-1, +/-2, +/-3, +/-4, +/-6, +/-12
1
1
1
2
-5
-6
1
3
-2
3
-2
-8
-1
This is a
remainder,
therefore Not a
factor
1
1
2
-5
-6
-1
-1
6
1
-6
0
No remainder,
therefore a
factor
These are all
the possible
factors of the
polynomial
(x + 1)(x2 + x - 6)
(x + 1)(x + 3)(x - 2)
Example 2: Factor: x3 - x2 - 16x 20
p: +/-1, +/-2, +/-4, +/-5, +/-10, +/- 20
q: +/-1
p/q: +/-1, +/-2, +/-4, +/-5, +/-10, +/-20
1
1
1
-1
-16
-20
1
0
-16
0
-16
-36
-1 1
This is a
remainder,
therefore Not a
factor
1
-1
-16
-20 -2 1
-1
2
14
-2
-14
-6
1
These are all
the possible
factors of the
polynomial
-1 -16 -20
-2
6
20
-3
-10
0
This is a
No remainder,
remainder,
therefore a
therefore Not a
factor
factor
(x + 2)(x2 - 3x - 10)
(x + 2)(x + 2)(x - 5)
Example 3: Factor: x4 - 11x3 + 27x2 + 11x - 28
p: +/-1, +/-2, +/-4, +/-7, +/-14, +/- 28
These are all
the possible
factors of the
polynomial
q: +/-1
p/q: +/-1, +/-2, +/-4, +/-7, +/-14, +/-28
1
1
-11
1
1
-10
27
11
-28
-10 17
28
17
28
0
(x - 1)(x3 - 10x2 + 17x + 28)
(x - 1)(x + 1)(x2 - 11x + 28)
(x - 1)(x + 1)(x - 4)(x - 7)
-1
1
1
-10
17
28
-1
11
-28
-11
28
0
Worksheet: “Factoring: Synthetic Division”
Example 4: Factor: 12x5 - 46x4 + 50x3 - 12x2
2
6
6
-23
25
-6
12 -22
6
-11
3
0
2x2(6x3 - 23x2 + 25x – 6)
2(x - 2)(6x2 - 11x + 3)
2(x - 2)(3x - 1)(2x - 3)
p: +/-1, +/-2, +/-3, +/-6
q: +/-1, +/-2, +/-3, +/-6
P/q: +/-1, +/-1/2, +/-1/3,
+/-1/6, +/-2, +/-2/3,
+/-1/6, +/-3, +/-3/2, +/-6
Example 5: Factor: 12x7 - 12x6 - 3x5 - 321x4 + 324x3 + 81x2 - 81x
1
4
4
-4
-1
-107
4
0
0
-1 -108
-1
108
27
-27
-108
0
27
0
27
0
3x(4x6 - 4x5 – x4 - 107x3 + 108x2 + 27x - 27)
3x(x - 1)(4x5 - x3 – 108x2 + 27)
3x(x - 1)(4x2 - 1)(x3 - 27)
3x(x - 1)(2x + 1)(2x - 1)(x - 3)(x2 + 3x + 9)
Worksheet: “Factoring: Synthetic Division II”