## Factorisation

Factorisation of an algebraic expression is the reverse process of expanding.
It involves placing brackets into an expression.
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

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 3

6xy − 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)

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 x2 is 1.

e.g. Factorise x2 + 6x + 8

The x2 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 x2+ 6x + 8 = (x + 4)(x + 2)

2. If the coefficient of x2 is not 1.

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

e.g. 2x2 + 13x + 6 = (2x + 1)(x + 6)

The 2x and the x are multiplied to give 2x2

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

### Difference of Two Squares

(a + b)(a − b) = a2 − b2

also means a2 − b2 = (a + b)(a − b)

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

Examples

x2 − 9 = (x + 3)(x − 3)

81q2- 100p2 = (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. a2 + 2ab + b2 = (a + b)2

a2 − 2ab + b2 = (a − b)2

### Algebraic Fractions

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

e.g. 