Factorisation of an algebraic expression is the reverse process of expanding.

It involves placing brackets into an expression.

A **sum** of terms then becomes a **product**.

Once an expression has been factorised, it should then be expanded to check it is correct.

Some expressions cannot be factorised.

Common Factors |
Grouping |
Quadratics |
Perfect Squares |

Common Factors

If every term of the expression has a common factor, this common factor should be removed and placed outside a set of brackets.

e.g. 9p + 12 = 3 × 3p + 3 × 4 = 3(3p + 4)

the common factor is 36xy − 18x = 6x . y − 6x . 3 = 6x(y − 3)

the common factor is 6x

### Grouping

Sometimes terms need to be grouped together to find common factors.

e.g. ab + 2b + 5a + 10 = b(a + 2) + 5(a + 2) the common factor is (a + 2)

= (a + 2)(b + 5)

### Quadratics

The aim when factorising quadratics should be to be able to do them mentally.

A trial-and-error process is used, then checked by expanding.

There are basically two types of quadratics:

**1. If the coefficient of x ^{2} is 1.**

e.g. Factorise x

^{2}+ 6x + 8

The x^{2} term must come from an x in each bracket. (x + ... )(x + ....)

The + 8 must come from two numbers multiplied together.

i.e. + 8 multiplied by + 1

+4 multiplied by +2

or-8 multiplied by − 1-4 multiplied by -2

The + 6 must result from adding one of the above pairs of numbers.

The only correct two numbers would be + 4 and + 2.

**Therefore x ^{2}+ 6x + 8 = (x + 4)(x + 2)**

**2. If the coefficient of x ^{2} is not 1.**

A trial-and-error process is again used, with checking done by expanding.

e.g. 2x^{2} + 13x + 6 = (2x + 1)(x + 6)

The 2x and the x are multiplied to give 2x^{2}

The + 6 and the + 1 are multiplied to give + 6.

### Difference of Two Squares

(a + b)(a − b) = a^{2} − b^{2}

also means a^{2} − b^{2} = (a + b)(a − b)

This property helps to factorise expressions containing two squared terms that are being subtracted.

**Examples**

x

^{2}− 9 = (x + 3)(x − 3)81q

^{2}- 100p^{2}= (9q + 10p)(9q − 10p)

### Perfect Squares

The patterns of the perfect squares of the previous section on expanding, should be learned to assist with factorisation.

e.g. a^{2} + 2ab + b^{2} = (a + b)^{2}

a^{2} − 2ab + b^{2} = (a − b)^{2}

### Algebraic Fractions

Fractions involving algebraic expressions can often be simplified using factorising and cancelling.

e.g.