1. Chapter 8: Factoring

2. Chapter 8 : Factoring Fill in the titles on the foldable
Prime Factoring & Factor a monomial
Greatest Common Factor (GCF)
Factor Using Distributive Property
Factory by Grouping
Zero Product Property
Factoring Trinomials – x2 + bx + c
Factoring Trinomials – ax2 + bx + c
Factoring Differences of Squares
Factoring Perfect Squares
Square Root Property
Rational Expressions

3. 8.1 Prime factoring and factor a monomial (top)
Prime # = factors only include 1 and itself
Composite # = more than two factors
Ex: Prime factor 90
Prime numbers:
1, 3, 5, 7, 11, 13, 17, 23, 29, 31, 37 …. 90 2 45 3 15 3 5
2 x 3 x 3 x 5 =
2 x 32 x 5

4. 8.1 Prime factoring and factor a monomial (bottom)
Factor a monomial = list all factors separately (no exponents)
Ex: -66pq2
Ex: 38rs2t
-66 38 66 -1 2 19 2 33
2 x 19 x r x s x s x t 3 11
-1 x 2 x 3 x 11 x p x q x q

5. 8.1 Greatest Common Factor (top)
GCF = the largest factor that is in all the given monomials
1. factor all monomials
2. circle all common factors
3. Multiply all the circled factors

6. 8.1 Greatest Common Factor (bottom)
Ex: & 70
Ex: 36x2y & 54xy2z 36 84 70 54 2 18 2 42 2 35 2 27 2 9 2 21 5 7 3 9 3 3 3 7 3 3
2 x 2 x 3 x 7
2 x 2 x 3 x 3 x x x x x y
2 x 5 x 7
2 x 3 x 3 x 3 x x x y x y x z
2 x 7= 14
2 x 3 x 3 x x x y=
18xy

7. 8.2 Factor Using Distributive Property (top)
Find the GCF of the monomials
Write each term as a product of the GCF and the remaining factors
Simplify the remaining factors

8. 8.2 Factor Using Distributive Property (bottom)
Ex: 12a2 + 16a
Ex: 3p2q – 9pq2 + 36pq -9 36 12 16 -1 9 2 18 2 6 2 8 3 3 2 9 2 3 2 4 3 3 2 2
x 3 x p x p x q
2 x 2 x 3 x a x a
-1 x 3 x 3 x p x q x q
= 2 x 2 x a
=4a
3 x p x q =
3pq
2 x 2 x 2 x 2 x a
2 x 2 x 3 x 3 x p x q
4a(3a) +
4a(4) =
3pq(p) +
3pq(-3q) +
3pq(12) =
4a(3a + 4)
3pq(p - 3q + 12)

9. 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

10. 8.2 Factor by Grouping (bottom)
Ex: 4ab + 8b + 3a + 6
Ex: 3p – 2p2 – 18p + 27
(4ab + 8b)(+ 3a + 6)
(3p – 2p2 )( – 18p + 27)
-18 4 6 27 8 -1 18 2 2 2 3 3 9 2 4 2 9 3 3 2 2 3 3
2 x 2 x a x b
3 x a
3 x p
= 3
= p
-1 x 2 x 3 x 3 x p
=4b
2 x 2 x 2 x b
2 x 3
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)

11. 8.2 Zero Product Property (top)
Roots = the solutions to the equation
When an equation is factored and equal to zero:
Set each factor equal to zero and solve for the variable

12. 8.2 Zero Product Property (bottom)
Ex: 7f2 – 35f = 0
Ex: (d – 5)(3d + 4) = 0
-35
d – 5 = 0
3d + 4 = 0
7 x f x f
= 7f
-1 x 5 x 7 x f -1 35
3d = -4
d = 5 5 7
/3 /3
d = -4/3
7f(f)
+ 7f(-5)
7f(f – 5) = 0
Roots are d = 5 and -4/3
f – 5 = 0
7f = 0
/7 /7
f = 5
f = 0
Roots are f = 0 and 5

13. 8.3 Factoring Trinomials – x2 + bx + c (top)
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)
Foil to check your answer
Use Zero Product Property to solve if needed

14. 8.3 Factoring Trinomials – x2 + bx + c (bottom)
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)
FOIL
x2 + 2x + 4x + 8
x2 + 6x + 8
FOIL
s2 – 7s – 4s + 28
s2 – 11s + 28
FOIL
r2 – 6r + 4r - 24
r2 - 2x - 24
s – 4 = 0
s – 7 = 0
s = 4
s = 7
s = 4 and 7

15. 8.4 Factoring Trinomials – ax2 + bx + c (top)
Get everything on one side (equal to zero)
Put the first part in each set of parentheses
Find product of the first and last parts
Find the factors of the product
Fill in the pair of factors that adds or subtracts to the second part
Remove the GCF from one set of parentheses
Write what is left of the that group as one factor and then the other group as the other factor
if you can’t factor = prime
(use the zero product property to solve if needed)

16. 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
(5x )(5x )
1, 30
2, 15
3, 10
5, 6
5(2y )(2y )=0
(5x + 10)(5x + 3)
5(2y - 4)(2y - 3)=0
5x: x 5
= 5
2 x y x y
10: 2 5
= 2
2 x 2
(5x + 10) (x + 2)
(2y - 4) (y - 2)
(x + 2)(5x + 3)
5(y - 2)(2y - 3) = 0
Solve for y.
y – 2 = 0
2y – 3 = 0
y = 2 and 1.5

17. 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
Hint: find the gcf to pull it out and make the numbers smaller if possible
Ex: n2 - 25
Ex: 9x3 – 4x
n x n
5 x 5
x(9x2 – 4)
(n + 5)(n - 5)
x[ 3x x 3x
2 x 2]
x(3x + 2)(3x - 2)

18. 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)
0 = a(7a x 7a 11 x 11)
0 = a(7a + 11)(7a - 11)
x x x x x
-1 x x
= x2
= -1
3 x x x x
-1 x 3
a = 0
7a + 11 = 0
7a - 11 = 0
5[ x2(x + 3)
- 1(x + 3)]
7a = -11
7a = 11
/7 /7
/7 /7
5[(x2 – 1)(x + 3)]
5[(x x x 1 x 1)(x + 3)]
a= -11/7
a = 11/7
5(x + 1)(x - 1)(x + 3)
a = -11/7, 0, and 11/7

19. 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?
If any of these answers is no- it is not a perfect square trinomial

20. 8.6 Factoring Perfect Squares (bottom)
Ex: (x – 7)2
Ex: (a – 4)2
x2 – 14x + 49
a2 – 8a + 16
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

21. 8.7 Square Root Property Ex: (y – 8)2 = 7 Ex: (b – 7)2 = 36 b – 7 = 6
b = 13
b = 1

22. 11.2 Rational Expressions (top)

23. 11.2 Rational Expressions (bottom)