Factorisation

The reverse process of expansion is called factorisation. For example:

Factorise 2x+6x^{2}

2x+6x^{2}=2x(1+3x)

Always check that the factorised form obtained, when expanded, give the expression we started with.

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

= 2x + 6x^{2}

Let’s look at some examples.

Example

Factorise 5x^{3}-15xy

5x^{3}-15xy=5x(x^{2}-3y)


Example

Factorise the following:

i) 5x^{2}y+10x

ii) 4x^{2}-8x

iii) 8xy^{2}+12x^{2}y

iv) 15x^{2}y^{2}-25xy

v) 24x^{2}yz-8xy^{2}z

i) 5x^{2}y+10x=5x(xy+z)

ii) 4x^{2}-8x=4x(x-2)

iii) 8xy^{2}+12x^{2}y=4xy(2y+3x)

iv) 15x^{2}y^{2}-25xy=5xy(3xy-5)

v) 24x^{2}yz-8xy^{2}z=8xyz(3x-y)


Example

Factorise: 2xy+6x-3y-9

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

We can factorise this last expression further, since (y+3) appears twice.

So, 2xy+6x-3y-9=2x(y+3)-3(y+3)

=(y+3)(2x-3)

= (2x-3)(y+3)


Example

Factorise 2x^{2}-2x+5xy-5y

2x^{2}-2x+5xy-5y=2x(x-1)+5y(x-1)

= (2x+5y)(x-1)


Example

Factorise the following:

i) 6x^{2}+12xy

ii) 6x^{2}-12x^{2}y

iii) 4x^{3}+12x^{2}-6x

iv) 15x^{2}-20xy

v) 30xyz-20xy^{2}z

i) 6x^{2}+12xy=6x(x+2y)

ii) 6x^{2}-12x^{2}y=6x^{2}(1-2y)

iii) 4x^{3} + 12x^{2}-6x=2x(2x^{2}+6x-3)

iv) 15x^{2}-20xy=5x(3x-4y)

v) 30xyz-20xy^{2}z=10xyz(3-2y)