Factoring Polynomials by Grouping
Factoring by grouping is often used to factor a fourterm polynomial
such as 10x^{2} + 35x  6xy  21y.
Procedure â€”
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: 10x^{2} + 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. 
10x^{2} = 2
Â· 5 Â· x
Â· x
35x = 5 Â· 7
Â· x
6xy = 1 Â· 2 Â· 3 Â· x Â·
y 21y = 1 Â· 3 Â·
7 Â· y

Step 2 
Group terms
with common
factors. 
(10x^{2} + 35x) + (6xy  21y)
= (5x Â· 2x
+ 5x Â· 7)
+ (1 Â· 3y Â· 2x
+ (1 Â· 3y) Â· 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 10x^{2} + 35x  6xy  21y, we could
also group like this:
(10x^{2}  6xy) + (35x  21y)
2x(5x  3y) + 7(5x  3y)
(5x  3y)(2x + 7)
Example 2
Factor: 6x^{2 }+ 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. 
6x^{2} = 2
Â· 3 Â· x
Â· x
3xy = 3 Â· x
Â· y
2x = 2 Â· x = 1 Â· 2 Â· x y =
1 Â· y

Step 2 
Group terms
with common
factors. 
(6x^{2} + 3xy) + (2x + y)
= (3x Â· 2x
+ 3x Â·
y)
+ (1 Â· 2x
+ 1 Â· 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).
