Math Learning Center Supplement
698-1579
CB 116
Factor by Grouping
Factor by grouping simply uses multiple applications of factoring out the Greatest Common Factor
(GCF). To use factor by grouping there must be at least four terms. As always, the first step in factoring
is to see if all the terms have any common factors that can be factored out. Once the common factors (if
any) are factored out, we can then begin the process of factoring by grouping. Since the most common
application of factor by grouping is with four terms, our examples will only be done with four terms.
Step 1: Grouping the terms
Put parenthesis around the first two terms and then around the last two terms. If the third term has
a negative sign in front of it, include the negative sign inside the second set of parenthesis. Insert a “+”
sign between the two sets of parenthesis.
Examples:
Ungrouped:
Grouped:
15 x 2 + 9 x + 10 x + 6
Ungrouped:
Grouped:
21x 2 + 7 x − 6 x − 2
21x 2 + 7 x + (− 6 x − 2 )
(
Ungrouped:
Grouped:
9 x 2 + 6 x − 3x − 2
9 x 2 + 6 x + (− 3 x − 2 )
Ungrouped:
Grouped:
5 x 2 + 3x + 5 x + 2
5 x 2 + 3 x + (5 x + 2 )
(15x
(
(
2
)
+ 9 x + (10 x + 6 )
)
)
)
Step 2: Factoring out the GCF from each group
From each group, factor out the greatest common factor. If the first term in a group has a negative
sign, then factor out a negative. If the factors in each group have no common factor, then factor out a
one.
(
)
Ungrouped:
15 x 2 + 9 x + (10 x + 6 )
GCF factored out:
3 x(5 x + 3) + 2(5 x + 3)
Ungrouped:
GCF factored out:
Ungrouped:
GCF factored out:
(21x
)
+ 7 x + (− 6 x − 2 )
7 x(3 x + 1) − 2(3x + 1)
(9 x
2
)
+ 6 x + (− 3 x − 2 )
3 x(3 x + 2 ) − 1(3 x + 2 )
2
Math Learning Center Supplement
Ungrouped:
GCF factored out:
(5 x
698-1579
CB 116
)
+ 3 x + (5 x + 2 )
x(5 x + 3) + 1(5 x + 2 )
2
Step 3: Factor out the common binomial
Many students have a difficult time visualizing this step. Therefore let’s first look at an example
with monomial terms: 3x(y) + 2(y). The common factor is “y”. Therefore, if we factor out the “y”, we
get (y)(3x+2). If we use the same example and substitute (5x+3) for (y) we get 3x(5x+3) + 2(5x+3).
Factoring out the common binomial factor
(5x + 3) we get (5x +3)(3x + 2).
Note: Factoring by grouping does not always work. If there is not a common binomial factor, then you
may need to rearrange the terms. If it still does not factor, then it is prime.
More examples:
GCF factored out:
Common binomial factored out:
7 x(3 x + 1) − 2(3x + 1) (Make sure your binomials match)
(3x + 1)(7 x − 2)
GCF factored out:
Common binomial factored out:
3 x(3 x + 2 ) − 1(3 x + 2 ) (Make sure your binomials match)
(3x + 2)(3x − 1)
GCF factored out:
No common binomial factor:
x(5 x + 3) + 1(5 x + 2 )
Prime.
Examples: Start to finish
Ungrouped:
Grouped:
GCF factored out:
Common binomial factored out:
35 x 2 + 14 x + 5 x + 2
35 x 2 + 14 x + (5 x + 2 )
7 x(5 x + 2 ) + 1(5 x + 2 ) (Make sure your binomials match)
(5 x + 2)(7 x + 1)
Ungrouped:
Grouped:
GCF factored out:
Common binomial factored out:
56 x 2 − 8 x − 7 x + 1
56 x 2 − 8 x + (− 7 x + 1)
8 x(7 x − 1) − 1(7 x − 1) (Make sure your binomials match)
(7 x − 1)(8 x − 1)
(
(
)
)