# Factoring a Four Term Polynomial by Grouping

### Learning Outcomes

- Factor a four term polynomial by grouping terms

[latex]\begin{array}{l}\left(x+4\right)\left(x+2\right)\\=x^{2}+2x+4x+8\\=x^2+6x+8\end{array}[/latex]

Additionally, factoring by grouping is a technique that allows us to factor a polynomial whose terms don't all share a GCF. In the following example, we will introduce you to the technique. Remember, one of the main reasons to factor is because it will help solve polynomial equations.### Example

Factor [latex]a^2+3a+5a+15[/latex]Answer: There isn't a common factor between all four terms, so we will group the terms into pairs that will enable us to find a GCF for them. For example, we wouldn't want to group [latex]a^2\text{ and }15[/latex] because they don't share a common factor.

[latex]\left(a^2+3a\right)+\left(5a+15\right)[/latex]

Find the GCF of the first pair of terms.[latex]\begin{array}{l}\,\,\,\,a^2=a\cdot{a}\\\,\,\,\,3a=3\cdot{a}\\\text{GCF}=a\end{array}[/latex]

Factor the GCF,*a*, out of the first group.

[latex]\begin{array}{r}\left(a\cdot{a}+a\cdot{3}\right)+\left(5a+15\right)\\a\left(a+3\right)+\left(5a+15\right)\end{array}[/latex]

Find the GCF of the second pair of terms.[latex]\begin{array}{r}5a=5\cdot{a}\\15=5\cdot3\\\text{GCF}=5\,\,\,\,\,\,\,\end{array}[/latex]

Factor [latex]5[/latex] out of the second group.[latex]\begin{array}{l}a\left(a+3\right)+\left(5\cdot{a}+5\cdot3\right)\\a\left(a+3\right)+5\left(a+3\right)\end{array}[/latex]

Notice that the two terms have a common factor [latex]\left(a+3\right)[/latex].[latex]a\left(a+3\right)+5\left(a+3\right)[/latex]

Factor out the common factor [latex]\left(a+3\right)[/latex] from the two terms.[latex]\left(a+3\right)\left(a+5\right)[/latex]

Note how the a and 5 become a binomial sum, and the other factor. This is probably the most confusing part of factoring by grouping.#### Answer

[latex-display]a^2+3a+5a+15=\left(a+3\right)\left(a+5\right)[/latex-display]*grouping technique*. Broken down into individual steps, here's how to do it (you can also follow this process in the example below).

- Group the terms of the polynomial into pairs that share a GCF.
- Find the greatest common factor and then use the distributive property to pull out the GCF
- Look for the common binomial between the factored terms
- Factor the common binomial out of the groups, the other factors will make the other binomial

### Example

Factor [latex]2x^{2}+4x+5x+10[/latex].Answer: Group terms of the polynomial into pairs.

[latex]\left(2x^{2}+4x\right)+\left(5x+10\right)[/latex]

Factor out the like factor, [latex]2x[/latex], from the first group.[latex]2x\left(x+2\right)+\left(5x+10\right)[/latex]

Factor out the like factor, [latex]5[/latex], from the second group.[latex]2x\left(x+2\right)+5\left(x+2\right)[/latex]

Look for common factors between the factored forms of the paired terms. Here, the common factor is [latex](x+2)[/latex]. Factor out the common factor, [latex]\left(x+2\right)[/latex], from both terms.[latex]\left(x+2\right)\left(2x+5\right)[/latex]

The polynomial is now factored.#### Answer

[latex-display]\left(x+2\right)\left(2x+5\right)[/latex-display]### Example

Factor [latex]2x^{2}–3x+8x–12[/latex].Answer: Group terms into pairs.

[latex](2x^{2}–3x)+(8x–12)[/latex]

Factor the common factor, [latex]x[/latex], out of the first group and the common factor, [latex]4[/latex], out of the second group.[latex]x\left(2x–3\right)+4\left(2x–3\right)[/latex]

Factor out the common factor, [latex]\left(2x–3\right)[/latex], from both terms.[latex]\left(x+4\right)\left(2x–3\right)[/latex]

#### Answer

[latex-display]\left(x+4\right)\left(2x-3\right)[/latex-display]### Example

Factor [latex]3x^{2}+3x–2x–2[/latex].Answer: Group terms into pairs.

[latex]\left(3x^{2}+3x\right)+\left(-2x-2\right)[/latex]

Factor the common factor [latex]3x[/latex] out of first group.[latex]3x\left(x+1\right)+\left(-2x-2\right)[/latex]

Factor the common factor [latex]−2[/latex] out of the second group. Notice what happens to the signs within the parentheses once [latex]−2[/latex] is factored out.[latex]3x\left(x+1\right)-2\left(x+1\right)[/latex]

Factor out the common factor, [latex]\left(x+1\right)[/latex], from both terms.[latex]\left(x+1\right)\left(3x-2\right)[/latex]

#### Answer

[latex-display]\left(x+1\right)\left(3x-2\right)[/latex-display]### Example

Factor [latex]7x^{2}–21x+5x–5[/latex].Answer: Group terms into pairs.

[latex]\left(7x^{2}–21x\right)+\left(5x–5\right)[/latex]

Factor the common factor [latex]7x[/latex] out of the first group.[latex]7x\left(x-3\right)+\left(5x-5\right)[/latex]

Factor the common factor [latex]5[/latex] out of the second group.[latex]7x\left(x-3\right)+5\left(x-1\right)[/latex]

The two groups [latex]7x\left(x–3\right)[/latex] and [latex]5\left(x–1\right)[/latex] do not have any common factors, so this polynomial cannot be factored any further.[latex]7x\left(x–3\right)+5\left(x–1\right)[/latex]

#### Answer

Cannot be factored## Contribute!

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- Ex 2: Intro to Factor By Grouping Technique.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Ex 1: Intro to Factor By Grouping Technique Mathispower4u .
**Authored by:**James Sousa (Mathispower4u.com) .**License:**CC BY: Attribution. - Unit 12: Factoring, from Developmental Math: An Open Program.
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