5.4 Factoring
Greatest Common Factor,
Difference of Two Squares,
Grouping, and Trinomials
Factoring a polynomial
means expressing it as a
product of other
polynomials.
Factoring Method #1
Factoring polynomials with a
common monomial factor
(using GCF).
**Always look for a GCF before
using any other factoring
method.
Steps:
1. Find the greatest common factor (GCF)
•
Look for the number GCF, then look for variables in
common. The lowest exponent of each variable in
common will be part of the GCF.
2. Divide the polynomial by the
GCF. The quotient is the other
factor.
3. Express the polynomial as the
product of the quotient and GCF.
3
2
2
Ex: 6c d  12c d  3cd
GCF  3cd
STEP #1:
Step#2: Divide
3
2 2
(6c d 12c d  3cd)  3cd 
2
2c  4cd  1
The answer should look like this:
3
2
2
Ex: 6c d  12c d  3cd
2
 3cd(2c  4cd  1)
Factor these on your own
looking for a GCF.
1. 6x  3x  12x  3x2x
3
2
2. 5x  10x  35  5x
2
2
2
 x  4
 2x  7
3. 16x y z  8x y z  12xy z
3 4
2
2 3
3 2
 4xy z4x y  2xz  3yz
2
2
2
2
Factoring Method #2
Factoring polynomials that are
a difference of perfect squares.
•
DIFFERENCE OF PERFECT
SQUARES
When factoring using a difference
of squares, look for the following 3
things:
1. Only 2 terms
2. Minus sign between them
3. Both terms must be perfect
squares
•
If all 3 of the above are true, write 2 ( ), one with a +
sign and one with a – sign ( + ) ( - ) The terms in
each of the parentheses will be the square root of
each term.
A “Difference of Perfect
Squares” is a binomial (*for
2 terms only*) and it factors
like this:
2
2
a  b  (a  b)(a  b)
To factor, express each term as a
square of a monomial then apply
2
2
the rule... a  b  (a  b)(a  b)
2
Ex: x  16 
2
2
x 4 
(x  4)( x  4)
Here is another example:
1 2
x 81
49
2
1 
2
1
1

x

9
x

9
x

9

7
7

7 
Try these on your own:
1. x  121
 x  11x  11
2. 9y  169x
2
2
2
 3y 13x 3y 13x 
3. x  16  x  2x  2x2  4
Be careful!
4
Factoring Method #3: Factor By Grouping
FACTOR BY GROUPING
1. When polynomials contain four terms, it
is sometimes easier to group terms in
order to factor.
2. Your goal is to create a common factor.
3. You can also move terms around in the
polynomial to create a common factor.
4. Practice makes it easier to recognize
common factors.
Factoring By Grouping
1. Group the first two terms and
the last two terms by putting
parentheses around them.
2. Factor out the GCF from each group
so that both sets of parentheses
contain the same factors.
3. Factor our GCF again and write the
answer as the product of two
binomials.
Ex: b
3
2
 3b  4b  12
Step 1: Group
 b  3b  4b 12
3
2
Step 2: Factor out GCF from each group
 b b  3  4b  3
2
Step 3: Factor out GCF again
 b  3b  4
2
3
2
2x

16x

8x

64
Ex2
3
2
:
 2 x  8x  4x 32


 2x  8x  4x  32
 2x x  8  4x  8
 2x 8x  4 
3
2
2
2
 2x 8x  2x  2
Factoring Chart
This chart will help you to determine
which method of factoring to use.
Type
Number of Terms
1.
2.
3.
4.
GCF
Diff. Of Squares
Trinomials
Grouping
2 or more
2
3
4
Box Method
1. Make a box with four squares
2. Make sure that the terms of the trinomial are
in descending order.
3. Put the first term in the top left box.
4. Put the last term in the bottom right box.
5. Multiply those two terms together.
6. List factors of the product in #5 that will add
together to get the middle term.
7. Put those in the other two boxes.
8. Find the GCF of each row and column – that
is your trinomial factored. Only take out a
negative if the first box in the row or
column is a negative number.
Trinomials
x  7x  6
2
Trinomials
2a  3a  1
2
Trinomials
6c  13c  6
2
Trinomials
12m  m  6
2
1) Factor 2x2 + 9x + 10
(x + 2)(2x + 5)
2) Factor
2
6y
- 13y - 5
(2y - 5)(3y + 1)
3)
2
12x
+ 11x - 5
(4x + 5)(3x - 1)
4) 5x - 6 +
2
x
x2 + 5x - 6
(x - 1)(x + 6)