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 can be rewritten as using this process. We begin by rewriting the original expression as and then factor each portion of the expression to obtain . We then pull out the GCF of to find the factored expression.
A General Note: Factor by Grouping
To factor a trinomial in the form by grouping, we find two numbers with a product of and a sum of . We use these numbers to divide the 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 , factor by grouping.
- List factors of .
- Find and , a pair of factors of with a sum of .
- Rewrite the original expression as .
- Pull out the GCF of .
- Pull out the GCF of .
- Factor out the GCF of the expression.
Example 3: Factoring a Trinomial by Grouping
Factor by grouping.Solution
We have a trinomial with , and . First, determine . We need to find two numbers with a product of and a sum of . In the table, we list factors until we find a pair with the desired sum.Factors of | Sum of Factors |
---|---|
29 | |
13 | |
7 |
Analysis of the Solution
We can check our work by multiplying. Use FOIL to confirm that .Try It 3
Factor the following.a. b.
Solution