## 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…

\[{\purple (3x^2-11)(x^3+4)}\]