Factoring Polynomials by Grouping
Factoring by grouping is often used to factor a four-term polynomial
such as 10x2 + 35x - 6xy - 21y.
Procedure —
To Factor a Polynomial by Grouping
Step 1 Factor each term.
Step 2 Group terms with common factors.
Step 3 In each group, factor out the GCF.
Step 4 Factor out the GCF of the polynomial from Step 3.
Example 1
Factor: 10x2 + 35x - 6xy - 21y
Solution
| Step 1 |
Factor each term.
The GCF of the first two terms is 5x. The GCF of the second two terms is 3y. |
10x2 = 2
· 5 · x
· x
35x = 5 · 7
· x
-6xy = -1 · 2 · 3 · x ·
y -21y = -1 · 3 ·
7 · y
|
| Step 2 |
Group terms
with common
factors. |
(10x2 + 35x) + (-6xy - 21y)
= (5x · 2x
+ 5x · 7)
+ (-1 · 3y · 2x
+ (-1 · 3y) · 7) |
| Step 3 |
In each group,
factor out the
GCF. |
= 5x(2x + 7) + (-3y)(2x + 7) |
| Step 4 |
Factor out the GCF of the polynomial from Step 3.
The binomial (2x + 7) is common to both groups. |
= (2x + 7)(5x - 3y) |
So, the factorization is (2x + 7)(5x - 3y).Note:
Often there is more than one way to form
two groups of two factors so that each has
at least one common factor.
For 10x2 + 35x - 6xy - 21y, we could
also group like this:
(10x2 - 6xy) + (35x - 21y)
2x(5x - 3y) + 7(5x - 3y)
(5x - 3y)(2x + 7)
Example 2
Factor: 6x2 + 3xy + 2x + y.
Solution
| Step 1 |
Factor each term.
The GCF of the first two terms is 3x. The GCF of the second two terms is
1. |
6x2 = 2
· 3 · x
· x
3xy = 3 · x
· y
2x = 2 · x = 1 · 2 · x y =
1 · y
|
| Step 2 |
Group terms
with common
factors. |
(6x2 + 3xy) + (2x + y)
= (3x · 2x
+ 3x ·
y)
+ (1 · 2x
+ 1 · y) |
| Step 3 |
In each group,
factor out the
GCF. |
= 3x(2x + y) + 1(2x + y) |
| Step 4 |
Factor out the GCF of the polynomial from Step 3.
The binomial (2x + 7) is common to both groups. |
= (2x + y)(3x + 1) |
Thus, the factorization is (2x + y)(3x + 1). | 
