1. Chapter 8: Factoring

2. Chapter 8 : Factoring Fill in the titles on the foldable
Greatest Common Factor (GCF)
Grouping
Trinomials – x2 + bx + c
Trinomials – ax2 + bx + c
Differences of Squares
Perfect Squares
Sums and Differences of Cubes
Quadratic Form
Combinations

3. 8.2 Greatest Common Factor (top)
Find the GCF of the terms
Write the GCF then the remaining part of each term in parentheses

4. 8.2 Greatest Common Factor (bottom)
Ex: 12a2 + 16a
Ex: 3p2q – 9pq2 + 36pq
2 x 2 x 3 x a x a
x 3 x p x p x q
= 2 x 2 x a
=4a
-1 x 3 x 3 x p x q x q
2 x 2 x 2 x 2 x a
3 x p x q =
3pq
2 x 2 x 3 x 3 x p x q
4a(3a + 4)
3pq(p - 3q + 12)

5. Your Turn – What is the Greatest Common Factor (GCF)
Ex: 15x + 20
Ex: 8x³y + 2x²y² - 4 xy³
GCF=5
2 x x x y =
2xy
=5(3x + 4)
=2xy(4x² + xy – 2y²)

6. 8.2 Factor by Grouping (top)
Group the terms (first two and last two)
Find the GCF of each group
Write each group as a product of the GCF and the remaining factors
Combine the GCFs in a group and write the other group as the second factor

7. 8.2 Factor by Grouping (bottom)
Ex: 4ab + 8b + 3a + 6
Ex: 3p – 2p2 – 18p + 27
(4ab + 8b)( +3a + 6)
(3p – 2p2 )( – 18p + 27)
2 x 2 x a x b
3 x a
= 3
=4b
3 x p
2 x 2 x 2 x b
2 x 3
= p
-1 x 2 x 3 x 3 x p
-1 2 x p x p
= 9
3 x 3 x 3
4b(a + 2)
+3 (a + 2)
p(3 – 2p)
+ 9(-2p + 3)
(4b + 3)(a + 2)
(p + 9)(-2p + 3)

8. Your turn- Factor by Grouping
Ex: 5x(x – 2) + 6(x – 2)
Ex: 3x² - 2x + 6x – 4
(3x² – 2x )+( 6x - 4)
x(3x – 2)
+ 2(3x - 2)
(x – 2) (5x + 6)
(x + 2)(3x – 2)

9. Factoring Trinomials – x2 + bx + c
Get everything on one side (equal to zero)
Split into two groups ( )( ) = 0
Factor the first part x2 (x )(x ) = 0
Find all the factors of the third part (part c)
Fill in the factors of c that will add or subtract to make the second part (bx)
Use Distributive Property (Foil) to check your answer
Use Zero Product Property to solve if needed

10. Factoring Trinomials – x2 + bx + c
Ex: x2 + 6x + 8
Ex: r2 – 2r - 24
Ex: s2 – 11s + 28 = 0 8
1, 8
2, 4
(x )(x ) 24
1, 24
2, 12
3, 8
4, 6 28
1, 28
2, 14
4, 7
(r )(r )
(s )(s )
(x + 2)(x + 4)
(s- 4)(s - 7) = 0
(r + 4)(r - 6)
Check your work
Check your work
Check your work
FOIL
s2 – 7s – 4s + 28
s2 – 11s + 28
FOIL
x2 + 2x + 4x + 8
x2 + 6x + 8
FOIL
r2 – 6r + 4r - 24
r2 - 2x - 24
s – 4 = 0
s – 7 = 0
s = 4
s = 7
s = 4 and 7

11. 8.4 Factoring Trinomials – ax2 + bx + c (top)
Get everything on one side (equal to zero)
Find product of the first and last parts
Find the factors of the product
Rewrite the ax2 then fill in the pair of factors that adds or subtracts to make the second part followed by c
Factor by grouping
if you can’t factor = prime
(use the zero product property to solve if needed)

12. 8.4 Factoring Trinomials – ax2 + bx + c (bottom)
Hint: find the gcf to pull it out and make the numbers smaller if possible
Ex: 5x2 + 13x + 6
Ex: 10y2 - 35y + 30 = 0
2 x 6 = 12
5 x 6 = 30
5(2y2 - 7y + 6) = 0
1, 12
2, 6
3, 4
( ) ( )
5x2 + 3x + 10x + 6
1, 30
2, 15
3, 10
5, 6
5[(2y2 -3y)(-4y + 6)]=0
x(5x + 3) + 2(5x + 3)
5[y(2y - 3)-2(2y - 3)]=0
(x + 2)(5x + 3)
5(y - 2)(2y - 3) = 0
Solve for y.
y – 2 = 0
2y – 3 = 0
y = 2 and 1.5

13. 8.5 Factoring Differences of Squares (top)
Factor each term
Write one set of parentheses with the factors adding and one with the factors subtracting
Foil to check your answer
Ex: n2 - 25
n x n
5 x 5
(n + 5)(n - 5)

14. 8.5 Factoring Differences of Squares (bottom)
Ex: 5x3 + 15x2 – 5x - 15
Ex: 121a = 49a3
-121a a
5[x3 + 3x2 – x – 3]
0 = 49a3 – 121a
5[ (x3 + 3x2)( – x – 3)]
0 = a(49a2 – 121)
5[ x2(x + 3)
- 1(x + 3)]
0 = a(7a x 7a 11 x 11)
0 = a(7a + 11)(7a - 11)
5[(x2 – 1)(x + 3)]
a = 0
7a + 11 = 0
7a - 11 = 0
5[(x x x 1 x 1)(x + 3)]
7a = -11
7a = 11
5(x + 1)(x - 1)(x + 3)
/7 /7
/7 /7
a= -11/7
a = 11/7
a = -11/7, 0, and 11/7

15. 8.6 Factoring Perfect Squares (top)
Perfect Square Trinomial:
Is the first term a perfect square?
Is the last term a perfect square?
Does the second term = 2 x the product of the roots of the first and last terms?
The third term (c) must be positive
Use the sign of the second term
If any of these answers is no- it is not a perfect square trinomial

16. 8.6 Factoring Perfect Squares (bottom)
Ex:
x2 – 14x + 49
Ex:
a2 – 8a - 16
x x x
7 x 7
a x a
4 x 4
2 x x x 7= 14x
4 x 4 = 16 but it is a negative 16 so it can’t be a perfect square
(x – 7)2
Ex: 9y2 + 12y + 4
1. 9y2 =
3y x 3y
yes
2. 4 =
2 x 2
yes
3. 2(3y x 2) =
2(6y) =
12y
yes
(3y + 2)2

17. Sum and Difference of Cubes (top)
a3 + b3 = (a + b)(a2 – ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)
Same - Opposite - Always Positive
To remember the signs:
SOAP

18. Sum and Difference of Cubes (bottom)
Ex: x Ex: 27x3 – 64y3
Ex: 1000y3 – Ex: 125a3 + 27b3
x x x
3x 3x 3x 4y 4y 4y
x2 - 1x2x + 22
(3x - 4y)
(9x2 + 12xy + 16y2)
(x + 2)
(x2 – 2x + 4)
(10y - 6)
(100x2 + 60y + 36)
(5a + 3b)
(25a2 – 15ab + 9b2)

19. Quadratic Form (Bottom)
Ex: x4 + 3x Ex: x4 – 16
Ex: x Ex: x4 + 7x2 + 12
(x )(x )
(x )(x )
(x2 + 4)(x2 - 4)
(x2 + 1)(x2 + 2)
(x2 + 1)(x2 + 2)(x2 – 2)
( )( )
(x )(x )
(x2 + 3)(x2 + 4)

20. Combinations of Factoring Types (Top)
First look for the GCF and factor out if possible
Next look for patterns (perfect squares, difference of squares or sum and difference of cubes)
If no patterns appear factor the trinomial like normal

21. Combinations of Factoring Types (bottom)
Ex: 4x2 – Ex: 3x2 – 3x – 60
Ex: 8x6 – 64x3 Ex: 8x3 – 32x
4(x2 – 25)
3(x2 – x - 20) 20
1, 20
2, 10
4, 5
4(x + 5)(x – 5)
(x + 4)(x – 5)
8x3 (x3 – 8)
8x(x2 – 4)
8x3(x – 2)(x2 + 2x + 4)
8x(x + 2)(x – 2)