4.4 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 [latex]x[/latex] 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 [latex]2x^2+5x+3[/latex] can be rewritten as [latex]\left(2x+3\right)\left(x+1\right)[/latex] using this process. We begin by rewriting the original expression as [latex]2x^2+2x+3x+3[/latex] and then factor each portion of the expression to obtain [latex]2x\left(x+1\right)+3\left(x+1\right)[/latex]. We then pull out the GCF of [latex]\left(x+1\right)[/latex] to find the factored expression.
Factor by Grouping
To factor a trinomial in the form [latex]ax^2+bx+c[/latex] by grouping, we find two numbers with a product of [latex]ac[/latex] and a sum of [latex]b[/latex]. We use these numbers to divide the [latex]x[/latex] 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 [latex]ax^2+bx+c[/latex], factor by grouping.
- List factors of [latex]ac[/latex].
- Find [latex]p[/latex] and [latex]q[/latex], a pair of factors of [latex]ac[/latex] with a sum of [latex]b[/latex].
- Rewrite the original expression as [latex]ax^2+px+qx+c[/latex].
- Pull out the GCF of [latex]ax^2+px[/latex].
- Pull out the GCF of [latex]qx+c[/latex].
- Factor out the GCF of the expression.
Example 1: Factoring a Trinomial by Grouping
- Factor [latex]5x^2+7x−6[/latex] by grouping.
Analysis
We can check our work by multiplying. Use FOIL to confirm that [latex]\left(5x-3\right)\left(x+2\right)=5x^2+7x-6[/latex].
- Factor the following:
- [latex]2x^2+9x+9[/latex]
- [latex]6x^2+x− 1[/latex]
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