Factoring Polynomials by Grouping
Example
Factor: 20w^{2}  21y + 12w  35wy
Solution
To avoid an error with the signs, write each subtraction as an addition of
the opposite.
20w^{2}  21y + 12w  35wy
= 20w^{2} + (21y) + 12w + (35wy)
Step 1 Factor each term.
= 2 Â· 2 Â· 5
Â· w Â· w
+ (3 Â· 7 Â· y)
+ 2 Â· 2 Â· 3
Â· w + (5 Â· 7
Â· w Â· y)
Step 2 Group terms with common factors.
The first and third terms have common factors 2, 2, and w. Therefore,
group those terms together.
The second and fourth terms have common factors 7 and y. Therefore,
group those terms together.
= [2 Â· 2
Â· 5 Â· w
Â· w + 2 Â· 2
Â· 3 Â· w]
+ [(3 Â· 7
Â· y) + (5 Â·
7 Â· w Â· y)]
Step 3 In each group, factor out the GCF of the terms.
In the first group, factor out 2 Â· 2
Â· w.
= [2 Â· 2
Â· w(5 Â·
w + 3)] + [(3 Â· 7
Â· y) + (5 Â·
7 Â· w Â·
y)]
Simplify.
= [4w(5w + 3)] + [(3 Â· 7
Â· y) + (5 Â· 7
Â· w Â· y)]
In the second group, the common factor is 7y. However, both terms are
negative. Factor out 7y, rather than +7y, so the terms of the binomial
have the same sign as the binomial in the first group.
= [4w(5w + 3)] + [7y(3 + 5w)]
In the second group, write 3 + 5w as 5w + 3 to match the binomial in
the first group.
= [4w(5w + 3)] + [7y(5w + 3)]
Step 4 Factor out the GCF of the polynomial.
Factor out the binomial (5w + 3).
= (5w + 3)(4w  7y)
We have factored the given polynomial as follows:
20w^{2}  21y + 12w  35wy = (5w + 3)(4w  7y)
You can multiply to check the factorization. We leave the check to you.
Within the brackets, the third and fourth terms have no common factor
except 1. Therefore, group the third and fourth terms together and write
each with factor 1.
Step 3 In each group, factor out the GCF of the terms.
In the first group, factor out 2
Â· x.
In the second group, factor out 1. 
= 3w[2 Â·
x (x + 3 Â· y) + 1(x + 3
Â· y)] 
Step 4 Factor out the GCF of the polynomial.
Factor out (x + 3y). 
=3w[(x + 3y)(2x + 1)] 
We can multiply to check the factorization.
Is
Is
Is 
3w[(x + 3y)(2x + 1)]
3w[2x^{2} + x + 6xy + 3y]
6wx^{2} + 3wx + 18wxy + 9wy 
= 6wx^{2} + 18wxy + 3wx + 9wy ?
= 6wx^{2} + 18wxy + 3wx + 9wy ?
= 6wx^{2} + 18wxy + 3wx + 9wy ? Yes 
