What does it mean to factorise algebraic expressions?

Factorising means the breaking or the decomposition of any entity into the product of other entities.

Factorising is the process of finding the factors. It is like dividing an expression into a multiplication of relevant expressions. It is the reverse of expanding.

In the factorisation method, the algebraic or quadratic equations are reduced into the simplest form, where the equations are represented as the product of factors. 

For example, the factorisation of $$a^{2}-9$$ is $$(a-3)(a+3)$$. This means that both $$(a-3)$$ and $$(a+3)$$ are the factors of the expression $$a^{2}-9$$.

E2.2: Use brackets and extract the common factors 

There are three ways to factorise an algebraic expression. The most common methods used for the factorisation of algebraic expressions are discussed below:

Factorisation using common factors

To factorise an algebraic expression using common factors, you need to find the highest common factor in the given algebraic expression. Then, you have to group the term accordingly. You can also say that the opposite process of an expression is its factorisation.

For example, the factorisation of the expression $$2a^{2}-22a$$ will be $$2a(a-11)$$. This is because the highest common factor in the expression $$2a^{2}-22a$$ will be $$2a$$, taking common $$2a$$ from the whole expression and rearranging the remaining expression.

Factorisation by regrouping terms

In the expression, you can also find that every expression may not have a common factor. For that, you need to take the common factor separately.

Suppose you have an expression $$5x+y-yx-5$$. In this expression, nothing is common in all the terms. So, you need to make groups such that the first and the last terms of the expression include $$5$$ as common and the middle two terms include $$y$$ as common so that they can be rearranged as $$5x-5+y-yx=5(x-1)-y(x-1)$$. After taking the common, you will receive an expression similar to $$5(x-1)-y(x-1)$$. Here, you can notice that $$x-1$$ is common in the whole expression. So taking $$x-1$$ common in the whole expression as $$(x-1)(5-y)$$, the factors of the expression $$5x+y-yx-5$$ is $$(x-1)(5-y)$$.

Factorising using standard identities

Finding factors are different identities by which factors of the given expression can be derived easily. Well, identities are equality relations that hold for all the values of the variables. The different identities used are:

• $$(a+b)^{2}=a^{2}+b^{2}+2ab$$, where $$a$$ and $$b$$ are two variables.

• $$(a-b)^{2}=a^{2}+b^{2}-2ab$$, where $$a$$ and $$b$$ are two variables.

• $$a^{2}-b^{2}=(a-b)(a+b)$$, where $$a$$ and $$b$$ are two variables.

Worked examples of factorising

Example 1: Factorise the expression $$x^{2}+7x+10$$.

Step 1: Write the given information.

$$x^{2}+7x+10$$

Step 2: Use factorising by regrouping method.

Look for the numbers whose product is equal to $$10$$ and the sum is equal to $$7$$.

Step 3: Write the final factor.

$$x^{2}+7x+10=x^{2}+2x+5x+10$$

$$(x+2)(x+5)$$

So, $$(x+2)(x+5)$$ is the final factor of the expression $$x^{2}+7x+10$$.

Example 2: Find the greatest common factor of $$8x^{8}+4x^{5}-16x^{4}$$.

Step 1: Write the given information.

$$8x^{8}+4x^{5}-16x^{4}$$

Step 2: Find the highest common factor.

The highest common factor is $$4x^{4}$$ among $$8x^{8}$$, $$4x^{5}$$ and $$16x^{4}$$. 

Step 3: Take $$4x^{4}$$ as a common factor in the whole expression.

$$8x^{8}+4x^{5}-16x^{4}=4x^{4}(2x^{4}+x-4)$$.

Step 4: Write the greatest common factor.

Among $$4x^{4}(2x^{4}+x-4)$$, the greatest common factor is $$4x^{4}$$.

So, $$4x^{4}$$ is the greatest common factor of the expression $$8x^{8}+4x^{5}-16x^{4}$$.

Example 3: Factorise the expression $$x^{2}-961$$.

Step 1: Write the given information.

$$x^{2}-961$$

Step 2: Write the formula of standard identities for factorising.

$$a^{2}-b^{2}=(a-b)(a+b)$$

Step 3: Apply the above formula.

$$x^{2}-961=(x+31)(x-31)$$

So, $$(x+31)(x-31)$$ is the factor of the expression $$x^{2}-961$$.

Example 4: Factorise the expression $$12xy-8xz+4x^{2}y$$.

Step 1: Write the given information.

$$12xy-8xz+4x^{2}y$$

Step 2: Find the highest common factor.

The highest common factor is $$4x$$ among $$12xy$$, $$8xz$$ and $$4x^{2}y$$. 

Step 3: Take $$4x$$ as a common factor in the whole expression.

$$12xy-8xz+4x^{2}y=4x(3y-2z+xy)$$

So, $$4x(3y-2z+xy)$$ is the factor of the expression $$12xy-8xz+4x^{2}y$$.