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 $2{x}^{2}+5x+3$ can be rewritten as $\left(2x+3\right)\left(x+1\right)$ using this process. We begin by rewriting the original expression as $2{x}^{2}+2x+3x+3$ and then factor each portion of the expression to obtain $2x\left(x+1\right)+3\left(x+1\right)$ . We then pull out the GCF of $\left(x+1\right)$ to find the factored expression.

A General Note: Factor by Grouping

To factor a trinomial in the form $a{x}^{2}+bx+c$ 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 $a{x}^{2}+bx+c$ , factor by grouping.

List factors of $ac$ .
Find $p$ and $q$ , a pair of factors of $ac$ with a sum of $b$ .
Rewrite the original expression as $a{x}^{2}+px+qx+c$ .
Pull out the GCF of $a{x}^{2}+px$ .
Pull out the GCF of $qx+c$ .
Factor out the GCF of the expression.

Example 3: Factoring a Trinomial by Grouping

Factor $5{x}^{2}+7x - 6$ 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 $q=10$ .

\begin{array}{cc} 5 x^{2} - 3 x + 10 x - 6 & Rewrite the original expression as a x^{2} + p x + q x + c . \\ x (5 x - 3) + 2 (5 x - 3) & Factor out the GCF of each part . \\ (5 x - 3) (x + 2) & Factor out the GCF of the expression . \end{array}

$\begin{array}{cc}5{x}^{2}-3x+10x - 6 \hfill & \text{Rewrite the original expression as }a{x}^{2}+px+qx+c.\hfill \\ x\left(5x - 3\right)+2\left(5x - 3\right)\hfill & \text{Factor out the GCF of each part}.\hfill \\ \left(5x - 3\right)\left(x+2\right)\hfill & \text{Factor out the GCF}\text{ }\text{ of the expression}.\hfill \end{array}$

Analysis of the Solution

We can check our work by multiplying. Use FOIL to confirm that $\left(5x - 3\right)\left(x+2\right)=5{x}^{2}+7x - 6$ .

Try It 3

Factor the following.

a. $2{x}^{2}+9x+9$
b. $6{x}^{2}+x - 1$

Solution

Factoring Polynomials

A General Note: Factor by Grouping

How To: Given a trinomial in the form $a{x}^{2}+bx+c$ , factor by grouping.

Example 3: Factoring a Trinomial by Grouping

Solution

Analysis of the Solution

Try It 3

Candela Citations

Factoring Polynomials