### GCSE Maths

Numbers
Algebra
Geometry and Measures
Probability
Statistics

# Factorisation

Factorisation is the reverse process of expansion. It involves breaking down a given algebraic expression into simpler components or factors. By doing so, it becomes easier to perform various calculations and solve mathematical problems.

Let’s look at an example to understand the concept of factorisation:

Factorise 2x + 6x²

2x + 6x² = 2x(1 + 3x)

To ensure the factorisation is correct, expand the factorised form and compare it with the original expression:

2x(1 + 3x) = 2x(1) + 2x(3x)

= 2x + 6x²

Now, let’s look at some more examples.

## Examples

Example 1:

Factorise 5x³ – 15xy

5x³ – 15xy = 5x(x² – 3y)

Example 2:

Factorise the following expressions:

i) 5x²y + 10x

ii) 4x² – 8x

iii) 8xy² + 12x²y

iv) 15x²y² – 25xy

v) 24x²yz – 8xy²z

i) 5x²y + 10x = 5x(xy + 2)

ii) 4x² – 8x = 4x(x – 2)

iii) 8xy² + 12x²y = 4xy(2y + 3x)

iv) 15x²y² – 25xy = 5xy(3xy – 5)

v) 24x²yz – 8xy²z = 8xyz(3x – y)

Example 3:

Factorise 2xy + 6x – 3y – 9

2xy + 6x – 3y – 9 = 2x(y + 3) – 3(y + 3)

This expression can be factorised further since (y + 3) appears twice:

2xy + 6x – 3y – 9 = (2x – 3)(y + 3)

Example 4:

Factorise 2x² – 2x + 5xy – 5y

2x² – 2x + 5xy – 5y = 2x(x – 1) + 5y(x – 1)

= (2x + 5y)(x – 1)

Example 5:

Factorise the following expressions:

i) 6x² + 12xy

ii) 6x² – 12x²y

iii) 4x³ + 12x² – 6x

iv) 15x² – 20xy

v) 30xyz – 20xy²z

i) 6x² + 12xy = 6x(x + 2y)

ii) 6x² – 12x²y = 6x²(1 – 2y)

iii) 4x³ + 12x² – 6x = 2x(2x² + 6x – 3)

iv) 15x² – 20xy = 5x(3x – 4y)

v) 30xyz – 20xy²z = 10xyz(3 – 2y)