Factorising

​​​​In a nutshell

Factorising is the inverse process of expanding brackets. Factorising helps you to find the roots of equations and they offer an alternative, sometimes more useful, way to write expressions. 

Factorising a linear expression

To factorise a linear expression, find all common factors between each term and factor them out. As a result, divide each term by these factors. Write the result inside brackets, with the factored out factors sitting before the brackets.

Example 1

Factorise fully $$4x+32$$.

There is a common factor of $$4$$ between these terms. Hence:

​$$4x+32=\underline{4(x+8)}$$​​

This method can work with non-linear expressions.

Example 2

Factorise fully $$5x^3y^2-10xy^3+20xy^2$$.

Each term has a common factor of $$5xy^2$$. Hence the factorised form is:

​$$5x^3y^2-10xy^3+20xy^2=\underline{5xy^2(x^2-2y+4)}$$​​

This is fully factorised since there are no more common factors between the terms.

Factorising a quadratic expression

PROCEDURE

Example 3

Factorise fully $$x^2+12x+20$$.

You have that $$a=1$$, $$b=12$$ and $$c=20$$. Set up the factorisation:

$$x^2+12x+20=(Ax+B)(Cx+D)$$​​

Now seek a $$B$$ and $$D$$ such that $$B+D=12$$ and $$BD=20$$. You can use $$B=2$$ and $$D=10$$:

$$x^2+12x+20=\underline{(x+2)(x+10)}$$​​

This is the fully factorised form.

There are some notable product that can help with factorisation:

• ​$$(a+b)^2 = a^2 + 2ab + b^2$$
• ​$$(a-b)^2 = a^2 -2ab + b^2$$​​
• ​$$(a+b)(a-b) = a^2 - b^2$$

Example 4

Factorise this expression as far as possible: $$4x^2 - 9y^2$$.

This is similar to one of the notable products: $$a^2 - b^2 = (a+b)(a-b)$$. Equate $$a ^2= 4x^2$$ and $$b^2 = 9y^2$$, therefore $$a = 2x$$ and $$b = 3y$$. 

Now write your factorised expression:

​$$4x^2-9y^2 =\underline{(2x + 3y)(2x-3y)}$$​​

Factorising simple cubic expressions

Some cubic expressions can be factorised partially, leaving an expression you already know how to factorise.

Example 5

Factorise this expression as far as possible: $$x^3 + 3x^2 - 10x$$.

First, extract the common factor $$x$$.

$$x^3 + 3x^2 -10x = x(x^2 + 3x -10)$$​​

Now you have a quadratic expression. Factorising this can be done by looking for two numbers that add to three and multiply to minus ten. The numbers minus five and two do this:

Therefore $$x^3 + 3x^2 -10 = \underline{x(x-5)(x+2)}$$​