Factoring by Grouping

A method of grouping the terms for taking out the common factor from the terms in an expression is called the factorization (or factorisation) by grouping.

Introduction

A polynomial may consist of two or more terms for representing an indeterminant quantity in mathematical form. In some cases, two or more terms in the expression have a factor commonly. So, they are arranged as a group and it is useful to take out the common factor from them. Hence, the method of factorisation (or factorization) is called the method of factoring by grouping.

Steps

There are three steps involved in factoring a polynomial by grouping.

Arrange the terms closer by identifying the common factors.
Group the common factor terms with parentheses (round brackets) and then express each term in every group in factor form.
Take out the factor common from all the groups.

Example

Let us learn how to factorise (or factorize) an expression by grouping the terms from the following example.

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

Group the Terms as per common factor

A polynomial can be given in any order but it is essential write the terms in order by identifying the common factor. The common factor in terms can be identified by comparing every term with another term. In this example, the terms $x^2y$ and $3xy$ have $xy$ as a common factor. Hence, write those two terms closely one after one. Similarly, the terms $3x$ and 9 have 3 as a common factor and write them one after one.

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

Now, represent each group terms by writing the terms inside the parentheses (round brackets).

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

Factorize the terms in expression

Now, factorise each term in every group by writing the terms in factor form.

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

Take out factor common

In first group, $xy$ is a common factor and it can be taken out common from the terms as per inverse operation of the distributive property. Similarly, $3$ is a common factor and it can also be taken out common from the second group by using the same principle.

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

Now, $x+3$ is a common factor in both terms. Hence, take out the factor $x+3$ common from the terms to complete the factoring process.

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