What Is Factorising?

Factorising is the opposite of muliplying out brackets. For example, if we multiply out
2x(x + 5)
we get
2x2 + 10x
So if the question is
Factorise 2x2 + 10x
then the answer is
2x(x + 5)

How To Do Factorising

When you’re asked to factorise an expression, the first thing to do is to look for the highest common factor of the numbers. Put that outside the brackets and divide the other numbers by it. For example:
12x + 9
3 ( 4x + 3 )

Now we need to do the same for each letter in turn. When looking for common factors of a letter, it’s just the highest amount of that letter that appears in the brackets. For example, x2 and x have a highest common factor of x. x3 and x5 have a common factor of x3. So:
15x + 10x2
5 ( 3x + 2x2 )
5x ( 3 + 2x )

We repeat this for each letter:
20xy4 - 16x3y7
4 ( 5xy4 - 4x3y7 )
4x ( 5y4 - 4x2y7 )
4xy4 ( 5 - 4x2y3 )

Sometimes a letter will appear in one part of the expression but not the other. In this case, we just leave it as it is:
6xy2 + 12y
6 ( xy2 + 2y )
6y ( xy + 2 )

Difference of Two Squares

‘Difference of two squares’ is a special factorisation that comes up a lot as a National 5 question. The pattern is:
a2 - b2 = ( a + b ) ( a - b )
Note that it only works if the original has “-“, not if it’s “+”.

Usually one of the things is a letter, the other is a square number, like
x2 - 16 = x2 - 42 = ( x + 2 ) ( x - 2 )

To make it more difficult, they can put a square number and a letter together:
9x2 - 25 = 32x2 - 52 = ( 3x + 5 ) ( 3x - 5)

If they want to ‘hide’ difference of two squares, they’ll multiply both terms by a number. Then you need to use ordinary highest factor factorising first. For example:
12x2 - 27 = 3 ( 4x2 - 9 ) = 3 ( 2x + 3 ) ( 2x - 3 )

Factorising and Cancelling

A common exam question requires you to factorise the top and/or bottom of an algebraic fraction and then cancel parts out. This sort of question doesn’t normally tell you that you need to factorise, you need to spot it. For example:
Simplify fully:
14x - 28
4x2 - 16

7 ( 2x - 4 )
4x2 - 16

   7 ( 2x - 4 )   
( 2x + 4 ) ( 2x - 4)

    7   
( 2x + 4 )