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]- 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 factoredContribute!
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CC licensed content, Shared previously
- 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. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.