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 \(ax^2+bx+c\), 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).

\[x^2 - 9 = (x + 3)(x - 3)\]