Common factor

To factorise an expression, rewrite it as a product of factors.

Thinking back to removing brackets, the answer is now the question and the question is now the answer.

We could ask the question, 'What was it before the brackets were removed?'

A good idea when factorising is to multiply out the brackets once you've got an answer to find that the answer matches with the question.

If it doesn't, then you'll know there's something wrong.


Factorise \(10 + 4x\)

The first thing to do is find the highest common factor (H.C.F) of \(10\) and \(4x\).

This will tell us the term that will go outside the bracket.

The Highest Common Factor (H.C.F.) = 2.

\[10 + 4x = 2(... + ...)\]

To get the terms inside the bracket, find \(2 \times ? = 10\) and then \(2 \times ? = 4x\).

This is \(5\) and \(2x\) respectively:

So \(10+4x=2(5+2x)\)

Remember to multiply out the brackets now to check that the answer is correct.

Now try the example questions below.


Factorise \(6a - 9\)

Both \(6a\) and \(9\) can be divided by 3.

Therefore the Highest Common Factor (H.C.F.) = 3.

\[6a - 9 = 3(2a - 3)\]


Factorise \(15 + 10x\)

\[= 5(3 + 2x)\]


Factorise \(x^{2}+5x\)

The common factor is \(x\)



Factorise \(3{a^2} - 12a\)

The common factor is 3a.

\[= 3a(a - 4)\]


Factorise \(20xy - 6x\)

\[= 2x(10y - 3)\]


Factorise \(20{y^2} - 5y\)

\[= 5y(4y - 1)\]