## Factorising Expressions

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. Always look for a common factor first.

e.g. 8x + 16 = 8(x + 2) the common factor is 8

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

The highest or largest common factor must be taken out. e.g. If you only take out a 6 from the example above the expression will not be fully factorised.

### 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.
 +8 multiplied by +1 OR +4 multiplied by +2 OR -8 multiplied by -1 OR -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.
• The middle term comes from 2x . 6 and 1 . x which add to give 13x.
Practice makes Perfect!

### Difference of Two Squares

a2 − b2 = (a + b)(a − b)

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

 Example Factorised 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

 Example Factorised x2 − 4x + 4 = (x − 2)(x − 2) = (x − 2)2 4x4 + 12x2 + 9 = (2x2 + 3)(2x2 + 3)