# Factorisation

Each term in an expression is formed by the product of factors. For example, in the expression, the term has been formed by the factors 2, and

and are prime factors of. In algebraic expressions we use 'irreducible' instead of 'prime' we say that is the irreducible form of

Consider the expression .

It can be expressed in its irreducible factor form as

The process of writing an algebraic expression as the product of two or more algebraic expressions is called factorisation.

Expression like are already in factor form.

Consider the expressions

.

To find the factors for the above expressions we need to follow systematic methods.

(i) Method of common factors:

Factorise:

We shall write each term as a product of irreducible factors.

3 is the common factor for both the terms.

By distributive law,

The factors are 3 and These factors are irreducible.

Factorise:

By distributive law,

Example: Factorise

Solution:

Example

Factorise

Solution

(ii) Factorisation by regrouping terms:

The terms in an expression are grouped in such a way that we can find a common factor in each group. When we do in this manner, we can find a common factor between the groups which leads to the factorization of the given expression. This method of finding the factors for the given expression is called regrouping.

Consider the expression Note that first two terms has common factors as 5 and

y. Last two terms have common factor 7. But there is no single factor common to all the terms.

. The expression is now in the form of product of factors. Its factors are . These factors are irreducible.

The same expression is considered but we are going to group the terms in the other way.

Suppose we regroup as. Then

Here we find that the factors are the same but in a different order.

In an expression, if we cannot find a common factor between the terms, then we can use the method of regrouping.

Example

Factorise

Solution