# Revision:Factorising

In order to look at factorising, we first have a reminder of what 'expanding' means.

## Expanding Brackets

Brackets should be expanded in the following ways: For an expression of the form:

,

the expanded version is:

,

i.e., multiply the term outside the bracket by everything inside the bracket. E.g. -

[remember ])

For an expression of the form:

,

the expanded version is:

,

in other words everything in the first bracket should be multiplied by everything in the second.

Expand :

Solution:

## Factorising

Factorising is the reverse of expanding brackets, so it is putting:

into the form:

.

This is an important way of solving quadratic equations.

The first step of factorising an expression is to 'take out' any common factors which the terms have. So if you were asked to factorise, since  goes into both terms, you would write:

.

There is no simple method of factorising a quadratic expression. One way, however, is as follows:

### Example

Factorise:.

[here the  has been split up into two numbers whose multiple is 36. 36 was chosen because this is the product of 12 and 3, the other two numbers].

The first two terms,  and  both divide by , so 'take out' this factor of :

[we can do this because ].

Now, make the last two expressions look like the expression in the bracket:

### Example

Factorise:.

We need to split the  into two numbers which multiply to give -8. This has to be 4 and -2:

Once you work out what is going on, this method makes factorising any expression easy. It is worth studying these examples further if you do not understand what is happening. Unfortunately, the only other method of factorising is by trial and error.

## The Difference of Two Squares

If you are asked to factorise an expression which is one square number minus another, you can factorise it immediately. This is because:

.

Factorise:.

[imagine that ].