## Factorising

Factorising and expanding are to extremely important skills that you should be comfortable using. Once you get used to the various methods, completing these questions will come as second nature.

### Expanding Brackets

Brackets are a way of making equations easier to write and understand. When you have brackets in an equation, the operations inside are completed first. For example:

(1 + 2) x 3 = 9

but

1 + (2 x 3) = 7

For more on order of operations see our section on BODMAS

In order to expand a bracket, you multiply everything outside the bracket by what’s inside, for example:

a(b + c) = ab + ac

Another example is:

Expand $y(3y + 5)$

Multiply everything inside the brackets by the $y$ outside.

$= (3y \times y) + (5 \times y)$

Remembering that $y \times y = y^2$

$= 3y^2 + 5y$

For two brackets multiplied together, this this a little bit more complicated, but the same principle applies.

Expand (a + b)(c + d)

Everything in the second set of brackets needs to me multiplied by everything in the first set.

So (a + b)(c + d) = ac + ad + bc + bd

Another example:

Expand $(3y + 7)(y - 3)$

Multiplying everything in the first set of brackets by the second set of brackets:

$= 3y(y) + 3y(-3) + 7(y) + 7(-3)$

Simplifying a little

$= 3y^2 - 9y + 7y - 21$

Simplifying some more

$= 3y^2 - 2y - 21$

Hence $(3y + 7)(y - 3) = 3y^2 - 2y - 21$

### Factorising

Factorising is essentially the opposite of expanding brackets. For example, it could involve putting an equation such as $3y^2 - 2y - 21$ into the form $(3y + 7)(y - 3)$.

In some basic factorising questions all you might need to do is remove a common factor, for example:

$x^2-5x = x(x-5)$ because every term contains at least one $x$

At first factorising Quadratic Equations can seem a little challenging, but once you have practiced a few it will soon seem simple. I promise!

We will go through the systematic process of factorising a Quadratic Equation.

Example – Factorise $x^2 + 2x - 15$

Because there’s just a 1 in front of the $x^2$, we know that the the answer will be in the form $(x + a)(x + b)$.

Now looking at our original equation, we need two numbers that multiply together to give us -15, and add together to give 2.

We now list the factors of -15, which are, (-1,15), (1,-15), (-3,5), (-5,3).

Looking at these the only two that add together to give 2 are (-3,5).

Hence our answer is $(x + 5)(x - 3)$.

The great thing about factorising quadratics is that we can quickly expand the brackets to see if we have got the correct answer.

It’s important to practice these as much as you can because factorising quadratics is an extremely important and useful skill in Mathematics.

The Difference of Two Squares

The Difference of Two Squares is a special kind of Quadratic Equations, and spotting these and knowing how to factorise can save you a lot of time and effort in an exam.

If you’re ever asked to factorise an equation which is one square minus another, then you can factorise this like such:

$x^2 - y^2 = (x + y)(x - y)$

If you expand the right hand side you see you get:

$(x + y)(x - y) = x^2 + xy - xy - y^2 = x^2 - y^2$.

Example – Factorise $x^2 - 4$

Notice that $4 = 2^2$, so following the steps above we get:

$x^2 - 4 = (x + 2)(x - 2)$

As with any factorising question, if you want to check that you’re right all you need to do is to expand the brackets and make sure that your answer is the same as the one given in the question.