Factorising

Factorising is the opposite of expanding, where we rewrite an expression as a product of its factors. The goal is to find common factors and express the original expression as a product.

Factorising Single Terms

When factorising an expression with a common factor, find the highest common factor (HCF) of all terms and factor it out.

$$6x^2+12xy$$

Both terms are both divisible by 6 and x. Our HCF is therefore 6x.

$$6x^2+12xy=6x(x+2y$$

Factorising Quadratics

To factorise a quadratic expression in the form $a{x}^2+bx+c$, look for two numbers that multiply to give $ac$ and add to give $b$.

$$x^2+5x+6$$

$$2+3=5$$

$$2\times 3=6$$

$$x^2+5x+6=(x+2)(x+3)$$

When a is greater than 1, it can be harder to factorise.

$$2x^2+7x+3$$

$$ac=2\times 3=6$$

$$1+6=7$$

$$1\times 6=6$$

Split the x term into 1x and 6x.

$$2x^2+7x+3=2x^2+x+6x+3$$

Now factorise the first pair of terms and second pair of terms.

$$x(2x+1)+3(2x+1)$$

The terms outside the bracket create our first quadratic bracket, and the repeated bracket is our second.

$$(x+3)(2x+1)$$

Difference of Squares

A special case of factorisation is the difference of squares. These occur when there are no x terms (the x term is zero) and the number term is a negative.

$$x^2-9$$

Square root the terms and put a plus in one bracket and a minus in the other (so the x terms cancel).

$$x^2-9=(x+3)(x-3)$$