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Factoring Review
Greatest Common Factor
This is the distributive property backwards.
Example: Factor each polynomial by factoring out the greatest common factor (GCF)
a. 5ab + 10a
The GCF of 5ab and 10a is 5a.
5ab + 10a = 5a(b) + 5a(2) = 5a(b + 2)
5
12
b. y – y
The GCF of y5 and y12 is y5.
y5 – y12 = y5(1) – y5(y7) = y5(1 – y7)
5
2
c. -9a + 18a – 3a
The GCF of -9a5, 18a2, and -3a is 3a.
-9a5 + 18a2 – 3a = 3a(-3a4) + 3a(6a) + 3a(-1) = 3a(-3a4 + 6a - 1)
White Board Activity:
Practice: Factor each polynomial by factoring out the GCF.
a. 10y + 25
b. x4 – x9
5(2y + 5)
x4(1 – x5)
c. 4x3 + 12x
d. 6a3b + 3a3b2 + 9a2b4
4x(x2 + 3)
3a2b(2a + ab + 3b3)
Example: Factor 5(x + 3) + y(x + 3).
(x + 3) is the GCF.
(x + 3)(5 + y)
White Board Activity:
Practice: Factor 7(p + 2) + q(p + 2).
(p + 2)(7 + q)
Factoring by Grouping
Example: Factor by grouping.
a. 15x3 – 10x2 + 6x – 4
(15x3 – 10x2) + (6x – 4)
5x2(3x – 2) + 2(3x – 2)
(3x – 2)(5x2 + 2)
2
b. 3x + 4xy – 3x – 4y)
(3x2 + 4xy) + (-3x – 4y)
x(3x + 4y) + -1(3x + 4y)
(3x + 4y)(x – 1)
c. 3x3 – 2x – 9x2 + 6
(3x3 – 2x) + (-9x2 + 6)
x(3x2 – 2) + -3(3x2 – 2)
(3x2 – 2)(x – 3)
White Board Activity:
Practice: Factor by grouping.
a. 28x3 – 7x2 + 12x – 3
7x2(4x – 1) + 3(4x – 1)
(4x – 1)(7x2 + 3)
c. 3x2 + 4xy + 3x + 4y
x(3x + 4y) + 1(3x + 4y)
(3x + 4y)(x + 1)
b. 2xy + 5y2 – 4x – 10y
y(2x + 5y) – 2(2x + 5y)
(2x + 5y)(y – 2)
d. 4x3 + x – 20x2 – 5
x(4x + 1) – 5(4x + 1)
(4x + 1)(x – 5)
Factoring x2 + bx + c
This is FOIL multiplication backwards.
Example: Factor each of the following:
a. x2 + 7x + 12
12: 1(12) -1(-12) 2(6) -2(-6) 3(4) -3(-4)
(x + 3)(x + 4)
b. x2 – 12x + 35
35: 1(35) -1(-35) 5(7) -5(-7)
(x – 5)(x – 7)
c. x2 + 4x – 12
-12: -1(12) 1(-12) -2(6) 2(-6) -3(4) 3(-4)
(x – 2)(x + 6)
d. x2 – x – 42
-42: -1(42) 1(-42) -2(21) 2(-21) -3(14) 3(-14)
(x + 6)(x – 7)
White Board Activity:
Practice: Factor each of the following.
a. x2 + 12x + 20
20: 1(20), 2(10), 4(5)
(x + 2)(x + 10)
c. x2 + 5x – 36
-36: -1(36), -2(18), -3(12), -4(9)
1(-36), 2(-18), 3(-12), 4(-9)
(x – 4)(x + 9)
e. x2 – 23x + 22
22: -1(-22), -2(-11)
(x – 1)(x – 22)
g. x2 + 9xy + 14y2
14: 1(14), 2(7)
(x + 2y)(x + 7y)
-6(7)
6(-7)
b. x2 – 27x + 50
50: -1(-50), -2(-25), -5(-10)
(x – 2)(x – 25)
d. x2 + 2x – 48
-48: -1(48), -2(24), -3(16), -4(12), -6(8)
1(-48), 2(-24), 3(-16), 4(-12), 6(-8)
(x – 6)(x + 8)
f. x2 – 3x – 40
-40: -1(40), -2(20), -4(10), -5(8)
1(-40), 2(-20), 4(-10), 5(-8)
(x + 5)(x – 8)
h. a2 – 17ab + 30b2
30: -1(-30), -2(-15), -3(-5)
(a – 2b)(a – 15b)
Factoring ax2 + bx + c
There are multiple “tricks” for doing these problems and you may have a former teacher who has shown
you one of these “tricks”. You may use your favorite “trick” but I also want you to know the “math” way of
doing the problems.
Factoring ax2 + bx + c by Factoring by Grouping
Examples: Factor each of the following using grouping.
a. 8x2 – 14x + 5
8(5) = 40: 1(40)
2(20)
4(10)
5(8)
-1(-40) -2(-20) -4(-10) -5(-8)
8x2 – 4x – 10x + 5
(8x2 – 4x) + (-10x + 5)
4x(2x – 1) + -5(2x – 1)
(2x – 1)(4x – 5)
b. 6x2 – 2x – 20
2(3x2 – x – 10)
3(10) = 30: -1(30) -2(15) -3(10) -5(6)
1(-30) 2(-15) 3(-10) 5(-6)
2
2(3x + 5x – 6x – 10)
2[(3x2 + 5x) + (-6x – 10)]
2[x(3x + 5) + -2(3x + 5)]
2(3x + 5)(x – 2)
c. 18y2 + 21y – 60
3(6y2 + 7y – 20)
6(20) = 120: -1(120) -2(60) -3(40) -4(30) -5(24) -6(20)
2(-60) 3(-40) 4(-30) 5(-24) 6(-20)
2
3(6y - 8y + 15y – 20)
3[(6y2 – 8y) + (15y – 20)]
3[2y(3y – 4) + 5(3y – 4)]
3(3y – 4)(2y + 5)
White Board Activity:
Practice: Factor each of the following using grouping.
a. 3x2 + 13x + 8
b.
2
3x : x(3x)
8: 1(8), 2(4)
(x + 3)(3x + 4)
b. 30x2 – 26x + 4
c.
2(15x2 – 13x + 2)
15x2: x(15x), 3x(5x)
2: 1(2)
2(5x – 1)(3x – 2)
-8(15)
8(-15)
12x2 + 19x + 5
12x2: x(12x), 2x(6x), 3x(4)
5: 1(5)
(3x + 1)(4x + 5)
6x2y – 7xy – 5y
y(6x2 – 7x – 5)
6x2: x(6x), 2x(3x)
5: 1(5)
y(2x + 1)(3x – 5)
One of the more popular “tricks” is called the “Slide and Divide” method.
Example: 1. Factor 3x2 + 11x + 6
a. “slide” the 3 to multiply by the 6.
x2 + 11x + 18
b. Foil factor the “slide” quadratic.
(x + 2)(x + 9)
c. Divide by the 3.
(x + 2/3)(x + 9/3) = (2x + 3)(x + 3)
2
2. Factor 8x – 22x + 5
a. x2 – 22x + 40
b. (x – 2)(x – 40)
c. (x – 2/8)(x – 40/8) = (x – ¼)(x – 5) = (4x – 1)(x – 5
-10(12)
10(-12)
3. 2x2 + 13x – 7
a. x2 + 13x – 14
b. (x + 14)(x – 1)
c. (x + 14/2)(x – ½) = (x + 7)(2x – 1)
White Board Activity:
Practice: Factor each of the following using “slide and divide”.
a. 5x2 + 27x + 10
b. 4x2 + 12x + 5
(5x + 2)(x + 5)
(2x + 1)(2x + 5)
2
2
c. 2x – 11x + 12
d. 6x – 5x + 1
(2x - 3)(x - 4)
(3x - 1)(2x – 1)
2
2
e. 3x + 14x – 5
f. 35x + 4x – 4
(3x – 1)(x + 5)
(5x + 2)(7x – 2)
g. 14x2 – 3xy – 2y2
h. 12a2 – 16ab – 3b2
(7x + 2y)(2x – y)
(6a + b)(2a – 3b)
Special Products
Example: Factor each of the following:
a. x2 + 8x + 16
16: 1(16) 2(8) 4(4)
(x + 4)(x + 4)
(x + 4)2
2
b. 4x + 12x + 9
4(9) = 36: 1(36) 2(18) 3(12) 4(9) 6(6)
4x2 + 6x + 6x + 9
(4x2 + 6x) + (6x + 9)
2x(2x + 3) + 3(2x + 3)
(2x + 3)(2x + 3)
(2x + 3)2
When the lead coefficient and the constant are both perfect squares, try the even split. If it
works you are done, if is doesn’t refer back to previous methods.
c. 9x2 – 12xy + 4y2
(3x – 2y)(3x – 2y)
(3x – 2y)2
Example: Factor each of the following:
a. z2 – 4
b. y2 – 25
c. 4x2 – 1
d. 25a2 – 9b2
(x – 2)(x + 2)
(y – 5)(y + 5)
(2x – 1)(2x + 1)
(5a – 3b)(5a + 3b)
White Board Activity:
Practice: Factor each of the following:
a. x2 + 4x + 4
(x + 2)2
c. c2 – 9/25
(c – 3/5)(c + 3/5)
e. 25x2 + 80x + 64
(5x + 8)2
b. x2 – 9
(x – 3)(x + 3)
d. x2 – 4/5x + 4/25
(x – 2/5)2
f. 9b2 – 1
(3b – 1)(3b + 1)
g. 16x2 – 49y2
(4x – 7y)(4x + 7y)
Assessment:
Question student pairs.
Independent Practice:
Factoring Assignment.
h. 49x2 + 14x + 1
(7x + 1)2