Factorising Expressions

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
Difference of Two Squares
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. 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)

 

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 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)

Click here to practice factorising quadratics.

 

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)