1. Objectives The student will be able to:
Factor using the greatest common factor (GCF) and
Factor by Grouping

2. Review: What is the GCF of 25a2 and 15a?
Let’s go one step further…
1) FACTOR 25a2 + 15a.
Find the GCF and divide each term by the GCF
25a2 + 15a = 5a( ___ + ___ )
Check your answer by distributing. 5a 3

3. 2) Factor 18x2 - 12x3. Find the GCF 6x2 Divide each term by the GCF
18x2 - 12x3 = 6x2( ___ - ___ )
Check your answer by distributing. 3 2x

4. 3) Factor 28a2b + 56abc2. GCF = 28ab Divide each term by the GCF
28a2b + 56abc2 = 28ab ( ___ + ___ )
Check your answer by distributing.
28ab(a + 2c2) a
2c2

5. Factor 20x2 - 24xy
x(20 – 24y)
2x(10x – 12y)
4(5x2 – 6xy)
4x(5x – 6y)

6. 5) Factor 28a2 + 21b - 35b2c2 GCF = 7 Divide each term by the GCF
Check your answer by distributing.
7(4a2 + 3b – 5b2c2)
4a2 3b
5b2c2

7. Factor 16xy2 - 24y2z + 40y2 2y2(8x – 12z + 20) 4y2(4x – 6z + 10)
8xy2z(2 – 3 + 5)

8. Factor by Grouping
When polynomials contain four terms, it is sometimes easier to group like terms in order to factor.
Your goal is to create a common factor.
You can also move terms around in the polynomial to create a common factor.
Practice makes you better in recognizing common factors.

9. Factor by Grouping Example 1:
FACTOR: 3xy - 21y + 5x – 35
Factor the first two terms:
3xy - 21y = 3y (x – 7)
Factor the last two terms:
+ 5x - 35 = 5 (x – 7)
The green parentheses are the same so it’s the common factor
Now you have a common factor
(x - 7) (3y + 5)

10. Factor by Grouping Example 2:
FACTOR: 6mx – 4m + 3rx – 2r
Factor the first two terms:
6mx – 4m = 2m (3x - 2)
Factor the last two terms:
+ 3rx – 2r = r (3x - 2)
The green parentheses are the same so it’s the common factor
Now you have a common factor
(3x - 2) (2m + r)

11. Factor by Grouping Example 3:
FACTOR: 15x – 3xy + 4y –20
Factor the first two terms:
15x – 3xy = 3x (5 – y)
Factor the last two terms:
+ 4y –20 = 4 (y – 5)
The green parentheses are opposites so change the sign on the 4
- 4 (-y + 5) or – 4 (5 - y)
Now you have a common factor
(5 – y) (3x – 4)

12. Objective The student will be able to:
use the zero product property to solve equations

13. Zero Product Property If a • b = 0 then a = 0, b = 0,
or both a and b equal 0.

14. 1. Solve (x + 3)(x - 5) = 0
Using the Zero Product Property, you know that either x + 3 = 0 or x - 5 = 0
Solve each equation.
x = -3 or x = 5
{-3, 5}

15. 2. Solve (2a + 4)(a + 7) = 0 2a + 4 = 0 or a + 7 = 0 2a = -4 or a = -7
{-7, -2}

16. 3t + 5 = 0 or t - 3 = 0 3t = -5 or t = 3 t = -5/3 or t = 3 {-5/3, 3}
3. Solve (3t + 5)(t - 3) = 0
3t + 5 = 0 or t - 3 = 0
3t = -5 or t = 3
t = -5/3 or t = 3
{-5/3, 3}

17. Solve (y – 3)(2y + 6) = 0
{-3, 3}
{-3, 6}
{3, 6}
{3, -6}

18. 4 steps for solving a quadratic equation
Set the equation equal to 0.
Factor the equation.
Set each part equal to 0 and solve.
Check your answer on the calculator.
Set = 0
Factor
Split/Solve
Check

19. 4. Solve x2 - 11x = 0 GCF = x x(x - 11) = 0 x = 0 or x - 11 = 0
{0, 11}
Set = 0
Factor
Split/Solve
Check

20. Put it in descending order.
5. Solve. -24a +144 = -a2
Put it in descending order.
a2 - 24a = 0
(a - 12)2 = 0
a - 12 = 0
a = 12
{12}
Set = 0
Factor
Split/Solve
Check

21. 6. Solve 4m2 + 25 = 20m 4m2 - 20m + 25 = 0 (2m - 5)2 = 0 2m - 5 = 0
Set = 0
Factor
Split/Solve
Check

22. 7. Solve x3 + 2x2 = 15x x3 + 2x2 - 15x = 0 x(x2 + 2x - 15) = 0
x = 0 or x + 5 = 0 or x - 3 = 0
{0, -5, 3}
Set = 0
Factor
Split/Solve
Check

23. Solve a2 – 3a = 40
{-8, 5}
{-5, 8}
{-8, -5}
{5, 8}

24. The degree will tell you how many answers you have!
Solve 4r3 – 16r = 0
{-16, 4}
{-4, 16}
{0, 2}
{0, 4}
{-2, 0, 2}
The degree will tell you how many answers you have!

25. Find two consecutive integers whose product is 240.
Let n = 1st integer.
Let n + 1 = 2nd integer.
n(n + 1) = 240
n2 + n = 240
n2 + n – 240 = 0
(n – 15)(n + 16) = 0
Set = 0
Factor
Split/Solve
Check

26. The consecutive integers are
(n – 15)(n + 16) = 0
n – 15 = 0 or n + 16 = 0
n = 15 or n = -16
The consecutive integers are
15, 16 or -16, -15.