Free Homework Math Help: Factoring By Grouping

Factoring by Grouping

One of the easiest methods of factoring a polynomial is by grouping the terms together that might have common factors and then removing those common factors. This method of factoring does not always work, however. When it doesn't one of the other factoring methods must be used. Example:

x² + 3x + 2

Begin by splitting the middle term into two parts each having its own term:

x² + 2x + x + 2

Rearrange the terms so that the terms with common factors are close together. The terms 2x and 2 both have two as a common factor so put these terms next to one another.

x² + x + 2x + 2

Next, group the first two terms with parenthesis and the last two terms with parenthesis.

(x² + x) + (2x + 2)

Factor out the common factors from each group.

x(x+1) + 2(x+1)

The result is two terms with both now having the common factor of (x+1).

Once again, factor out the common factor and the result should be:

(x+2)(x+1)

Sometimes, finding a way of grouping the terms is not always so obvious. For example:

6x² + 25x + 14 (6x² + 6x) + (19x + 14)

This does not work because the last two terms don't have common factors. The same can be said for:

(6x² + 11x) + (14x + 14)

Now it is the first two terms that have no common factors other than x. When the common factors are factored out, the result is:

x(6x + 11) + 14(x + 1)

There are no more common factors and it appears that the polynomial can't be converted to factored form. For this polynomial it is better to use other methods.