Factoring by Grouping

Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial 2x2+5x+3 can be rewritten as (2x+3)(x+1) using this process. We begin by rewriting the original expression as 2x2+2x+3x+3 and then factor each portion of the expression to obtain 2x(x+1)+3(x+1). We then pull out the GCF of (x+1) to find the factored expression.

A General Note: Factor by Grouping

To factor a trinomial in the form ax2+bx+c by grouping, we find two numbers with a product of ac and a sum of b. We use these numbers to divide the x term into the sum of two terms and factor each portion of the expression separately, then factor out the GCF of the entire expression.

How To: Given a trinomial in the form ax2+bx+c, factor by grouping.

  1. List factors of ac.
  2. Find p and q, a pair of factors of ac with a sum of b.
  3. Rewrite the original expression as ax2+px+qx+c.
  4. Pull out the GCF of ax2+px.
  5. Pull out the GCF of qx+c.
  6. Factor out the GCF of the expression.

Example 3: Factoring a Trinomial by Grouping

Factor 5x2+7x−6 by grouping.

Solution

We have a trinomial with a=5,b=7, and c=−6. First, determine ac=−30. We need to find two numbers with a product of −30 and a sum of 7. In the table, we list factors until we find a pair with the desired sum.

Factors of −30 Sum of Factors
1,−30 −29
−1,30 29
2,−15 −13
−2,15 13
3,−10 −7
−3,10 7

So p=−3 and q=10.

5x2−3x+10x−6Rewrite the original expression as ax2+px+qx+c.x(5x−3)+2(5x−3)Factor out the GCF of each part.(5x−3)(x+2)Factor out the GCF  of the expression.

Analysis of the Solution

We can check our work by multiplying. Use FOIL to confirm that (5x−3)(x+2)=5x2+7x−6.

Try It 3

Factor the following.

a.2x2+9x+9
b. 6x2+x−1

Solution