Factoring Polynomials by Grouping

Factoring by grouping is often used to factor a four-term polynomial.

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 of the terms.

Step 4 Factor out the GCF of the polynomial.

To check the factorization, multiply the factors.

Example 1

Factor: 5wx + 20x + 9w + 36

Solution

Step 1 Factor each term.

5wx + 20x + 9w + 36 = 5 · w · x + 2 · 2 · 5 · x + 3 · 3 · w + 2 · 2 · 3 · 3

Step 2 Group terms with common factors.

There are several ways to form the groups. Typically, we group the first two terms and the last two terms, provided at least one of the groups has a common factor other than 1 or -1.

= (5 · w · x + 2 · 2 · 5 · x) + (3 · 3 · w + 2 · 2 · 3 · 3)

The terms in the first group have common factors 5 and x.

The terms in the second group have common factors 3 and 3.

= (5 · w · x + 2 · 2 · 5 · x) + (3 · 3 · w + 2 · 2 · 3 · 3)

Factor 5 · x out of the first group and 3 · 3 out of the second group.

= 5 · x(w + 2 · 2) + 3 · 3(w + 2 · 2)

=  5x(w + 4) + 9(w + 4)

Step 4 Factor out the GCF of the polynomial.

= (w + 4)(5x + 9)

Thus, 5wx + 20x + 9w + 36 = (w + 4)(5x + 9)

We can multiply to check the factorization.

Is (w + 4)(5x + 9) = 5wx + 20x + 9w + 36 ?

Is 5wx + 9w + 20x + 36 = 5wx + 20x + 9w + 36 ? Yes