Factoring By Grouping

Factoring by Grouping

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.

(Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime.)

Example: How to Factor by Grouping

Factor: \(xy+3y+2x+6\).

Solution

This table gives the steps for factoring x y + 3 y + 2 x + 6. In the first row there is the statement, “group terms with common factors”. In the next column, there is the statement of no common factors of all 4 terms. The last column shows the first two terms grouped and the last two terms grouped.

The second row has the statement, “factor out the common factor from each group”. The second column in the second row states to factor out the GCF from the two separate groups. The third column in the second row has the expression y(x + 3) + 2(x + 3).

The third row has the statement, “factor the common factor from the expression”. The second column in this row points out there is a common factor of (x + 3). The third column in the third row shows the factor of (x + 3) factored from the two groups, (x + 3) times (y + 2).

The last row has the statement, “check”. The second column in this row states to multiply (x + 3)(y + 2). The product is shown in the last column of the original polynomial x y + 3 y + 2 x + 6.

Factor by grouping.

Group terms with common factors.
Factor out the common factor in each group.
Factor the common factor from the expression.
Check by multiplying the factors.

Example

Factor: \({x}^{2}+3x-2x-6\).

Solution

\(\begin{array}{cccc}\text{There is no GCF in all four terms.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{x}^{2}+3x\phantom{\rule{0.5em}{0ex}}-2x-6\hfill \\ \text{Separate into two parts.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\underset{└\_\_\_\_\_\_\_┘}{{x}^{2}+3x}\phantom{\rule{0.5em}{0ex}}\underset{└\_\_\_\_\_\_\_┘}{-2x-6}\hfill \\ \begin{array}{c}\text{Factor the GCF from both parts. Be careful}\hfill \\ \text{with the signs when factoring the GCF from}\hfill \\ \text{the last two terms.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\begin{array}{c}\hfill x\left(x+3\right)-2\left(x+3\right)\hfill \\ \hfill \left(x+3\right)\left(x-2\right)\hfill \end{array}\hfill \\ \text{Check on your own by multiplying.}\hfill & & & \end{array}\)

Optional Videos:

You can watch these videos below for additional instruction and practice with greatest common factors (GFCs) and factoring by grouping.

Greatest Common Factor (GCF)

Factoring Out the GCF of a Binomial

Greatest Common Factor (GCF) of Polynomials

This lesson is part of:

Factoring and Factorisation I

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