Factoring by Grouping

A method of factoring an expression (a polynomial) by grouping the terms to take out the common factor from them is called the factorization (or factorisation) by grouping.

Introduction

In mathematics, it is essential to simplify the expressions by factorization in some cases and it is only possible to factorize some polynomials by grouping. So, let’s learn how to factorise the expressions by grouping.

Steps

An expression can be factored by grouping in three simple following steps.

1. Arrange the terms of a polynomial in groups on the basis of common factors.
2. Factorize each group by taking out the common factor.
3. Take out the common factor from all the groups.

Remember, the common factors are taken out from the terms according to the distributive property.

Example

Factorize $9+3xy+x^2y+3x$

Step – 1

The polynomial contains four terms and observe them for common factors. The first and fourth terms contains a common factor $3$. Similarly, the second and third terms have $xy$ as a common factor. So, write all the terms in groups first.

$= \,\,\,$ $x^2y+3xy+3x+9$

$= \,\,\,$ $(x^2y+3xy)+(3x+9)$

Step – 2

Now, factorise each group by taking common factors out from the groups.

$= \,\,\,$ $xy(x+3)+3(x+3)$

Step – 3

The given algebraic expression now has two terms but they have $x+3$ as a common factor. So, take the common factor out from the groups.

$= \,\,\,$ $(x+3)(xy+3)$

Therefore, the algebraic expression $9+3xy+x^2y+3x$ is factored as $(x+3)(xy+3)$ by grouping in mathematics.