How to Factor by Grouping
- Split the polynomial into two groups.
- Factor out the GCF from each group individually.
- Make sure the remaining factors match and factor them out.
Examples
Factor by Grouping:
\[20x^2-45x+12x-27\]
How many terms are there?
There are four terms in the polynomial, so I can split it into two groups with two terms each.
\[{\yellow (}20x^2-45x{\yellow )}+{\yellow (}12x-27{\yellow )}\]
What is the GCF of each group?
\[{\yellow (}20x^2-45x{\yellow )}+{\yellow (}12x-27{\yellow )}\]
Both of the terms in the first group can be divided by \(\yellow 5x\). And both of the terms in the second group can be divided by \(\yellow 3\).
\[{\yellow 5x(}4x-9{\yellow )}+{\yellow 3(}4x-9{\yellow )}\]
Do the expressions inside the parentheses match?
Yes, the expression inside both sets of parentheses is \(4x+9\), so I can factor it out.
\[{\yellow (}4x-9{\yellow )(5x+3)}\]
The factored form of \(20x^2-45x+12x-27\) is…
\[\yellow (4x-9)(5x+3)\]
Factor by Grouping:
\[6x^3+21x^2-10x-35\]
How many terms are there?
There are four terms in the polynomial, so I can split it into two groups with two terms each.
\[{\green (}6x^3+21x^2{\green )}+{\green (}-10x-35{\green )}\]
What is the GCF of each group?
\[{\green (}6x^3+21x^2{\green )}+{\green (}-10x-35{\green )}\]
Both of the terms in the first group can be divided by \(\green 3x^2\). And both of the terms in the second group can be divided by \(\green -5\).
\[{\green 3x^2(}2x+7{\green )}{\green -5(}2x+7{\green )}\]
Do the expressions inside the parentheses match?
Yes, the expression inside both sets of parentheses is \(2x+7\), so I can factor it out.
\[{\green (}2x+7{\green )(3x^2-5)}\]
The factored form of \(6x^3+21x^2-10x-35\) is…
\[{\green (2x+7)(3x^2-5)}\]
Factor by Grouping:
\[3x^5-11x^3+12x^2-44\]
How many terms are there?
There are four terms in the polynomial, so I can split it into two groups with two terms each.
\[{\purple (}3x^5-11x^3{\purple )}+{\purple (}12x^2-44{\purple )}\]
What is the GCF of each group?
\[{\purple (}3x^5-11x^3{\purple )}+{\purple (}12x^2-44{\purple )}\]
Both of the terms in the first group can be divided by \(\purple x^3\). And both of the terms in the second group can be divided by \(\purple 4\).
\[{\purple x^3(}3x^2-11{\purple )}+{\purple 4(}3x^2-11{\purple )}\]
Do the expressions inside the parentheses match?
Yes, the expression inside both sets of parentheses is \(3x^2-11\), so I can factor it out.
\[{\purple (}3x^2-11{\purple )(x^3+4)}\]
The factored form of \(3x^5-11x^3+12x^2-44\) is…
\[{\purple (3x^2-11)(x^3+4)}\]
