- The document discusses factoring expressions using the greatest common factor (GCF) method and grouping method. - With the GCF method, the greatest common factor is determined for each term and factored out. - The grouping method is used when there are 4 or more terms. The first two terms are factored and the second two terms are factored, then the common factor is extracted. - Examples of applying both methods to factor polynomials are provided and worked through step-by-step.

1. Factoring Expressions
- Greatest Common Factor
(GCF)
- Grouping (4 terms)

2. Objectives
• I can factor expressions using
the Greatest Common Factor
Method (GCF)
• I can factor expressions using
the grouping method

3. Factoring?
• Factoring is a method to find the basic
numbers and variables that made up a
product.
• (Factor) x (Factor) = Product
• Some numbers are Prime, meaning they are
only divisible by themselves and 1

4. Method 1
• Greatest Common Factor (GCF) –
the greatest factor shared by two or
more numbers, monomials, or
polynomials
• ALWAYS try this factoring method
1st before any other method
• Divide Out the Biggest common
number/variable from each of the
terms

5. Greatest Common Factors
aka GCF’s
Find the GCF for each set of following numbers.
Find means tell what the terms have in common.
Hint: list the factors and find the greatest match.
a) 2, 6
b) -25, -40
c) 6, 18
d) 16, 32
e) 3, 8
2
-5
6
16
1
No common factors?
GCF =1

6. Greatest Common Factors
aka GCF’s
Find the GCF for each set of following numbers.
Hint: list the factors and find the greatest match.
a) x, x2
b) x2, x3
c) xy, x2y
d) 2x3, 8x2
e) 3x3, 6x2
f) 4x2, 5y3
x
x2
xy
2x2
3x2
1 No common factors?
GCF =1

7. Greatest Common Factors
aka GCF’s
Factor out the GCF for each polynomial:
Factor out means you need the GCF times the
remaining parts.
a) 2x + 4y
b) 5a – 5b
c) 18x – 6y
d) 2m + 6mn
e) 5x2y – 10xy
2(x + 2y)
5(a – b)
6(3x – y)
2m(1 + 3n)
How can you check?
5xy(x - 2)

8. FACTORING by GCF
Take out the GCF EX:
15xy2 – 10x3y + 25xy3
How:
Find what is in common
in each term and put in
front. See what is left
over.
Check answer by
distributing out.
Solution:
5xy( 3 y – 2 x 2 + 5 y 2 )

9. FACTORING
Take out the GCF EX:
2x4 – 8x3 + 4x2 – 6x
How:
Find what is in common
in each term and put in
front. See what is left
over.
Check answer by
distributing out.
Solution:
2x(x3 – 4x2 + 2x – 3)

10. Ex 1
•15x2 – 5x
•GCF = 5x
•5x(3x - 1)

11. Ex 2
•8x2 – x
•GCF = x
•x(8x - 1)

12. Factoring by Grouping
(When you have 4 terms or a higher even number)
Objective: After completing this section, students
should be able to factor polynomials by grouping.

13. Steps for factoring by grouping:
1. A polynomial must have 4 terms to factor by grouping.
ex. x3  x2  2x  2
2. We factor the first two terms and the second two terms
separately. Use the rules for GCF to factor these.
3 2 x  x 2x  2
The GCF of
x 3  x 2 is x 2
.
  2 x x 1 2x 1
3. Finally, we factor out the "common factor" from both terms.
This means we write the 1 term in front and the 2 terms
left over, +2 , in a separate set of parentheses.
   2 x 1 x  2
The GCF of
2x  2 is 2.
 
2
x
x


14. Examples:
1. 6x3 9x2  4x  6
3 2 6x 9x 4x  6
The GCF of
6x 3 9x 2 is 3x 2
.
The GCF of
  4x  6 is 2. 2 3x 2x 3 22x 3
These two terms must be the same.
   2  2x 3 3x  2
3 2 2. x  x  x 1
3 2 x  x x 1
The GCF of
x 3  x 2 is x 2
.
The GCF of
  x 1 is 1. 2 x x 1 1x 1
These two terms must be the same.
   2  x 1 x 1

15. Examples:
3. x3  2x2  x  2
3 2 x  2x x  2
The GCF of
x 3  2x 2 is x 2
.
The GCF of
  x  2 is 1. 2 x x 2 1x2
These two terms must be the same.    2  x  2 x 1
You must always check to see if the expression is factored completely. This
expression can still be factored using the rules for difference of two squares. (see 6.2)
   2  x  2 x 1
This is a difference of two squares.
 x2x1x1

16. Examples:
4. x2 y2  ay2  ab  bx2
2 2 2 2 x y  ay ab bx
The GCF of
x 2 y 2  ay 2 is y 2
. 2
The GCF of
  ab  bx is b. 2 2 y x  a   2 b a  x
These two terms must be the same.
You can rearrange the terms so that they are the same.
   2 2  y b x  a
3 2 5. x  x  2x  2
3 2 x  x 2x  2
The GCF of
x 3  x 2 is x 2
.
The GCF of
  2x  2 is 2. 2 x x 1 2x 1
These two terms must be the same.
But they are not the same. So this
polynomial is not factorable.
Not Factorable

17. Try These:
Factor by grouping.
3 2
x x x
a. 8  2  12 
3
3 2
x x x
b. 4  6  6 
9
3 2
x x x
c.   
1
2
a b a ab
d. 3  6  5 
10

18. Solutions: If you did not get these answers, click the green
button next to the solution to see it worked out.
   2

x x
a. 4 1 2 3
 
  2

x x
b. 2 3 2 3
 
    
x x x
c. 1 1 1
  
 a b  a

d. 2 3 5
 

19. BACK
a. 8x3  2x2 12x  3
3 2
x x x
x x x
8 2 12 3
2 4 1 3 4 1
  
  
   
  
2
2
x x
  
4 1 2 3
The GCF of
8x 3  2x 2 is 2x 2
.
The GCF of
12x 3 is 3.

20. BACK
b. 4x3  6x2  6x  9
3 2
x x x
x x x
4  6  6 
9
2 2  3  3 2 
3
   
  
2
2
x x
  
2 3 2 3
The GCF of
4x 3  6x 2 is 2x 2
.
The GCF of
6x  9 is 3.
When you factor a negative out of
a positive, you will get a negative.

21. BACK
3 2 c. 1 x x x   
3 2
x x x
x x x
  
1
  
   
2
1 1 1
   2

x x
  
1 1
     
x x x
   
1 1 1
The GCF of
x 3  x 2 is x 2
.
The GCF of
x 1 is 1.
Now factor the difference of squares.

22. BACK
d. 3a  6b  5a2 10ab
3 a  6 b  5 a 2 
10
ab
3 a  2 b  5 a a 
2
b
The GCF of
3a  6b is 3. 2
   
 a b  a

  
2 3 5
The GCF of
5a 10ab is 5a.