# ## Factor By Grouping

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## How to Factor by Grouping

1. Split the polynomial into two groups.
2. Factor out the GCF from each group individually.
3. 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…

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