Factorising – The Basics

In this video we are going to go over the basics of Factorising – what it is and how we do it.

We’re going to keep this video super short and sweet because we’ve already covered the polar opposite of Factorising. That is multiplying out brackets.

Factorising

When we multiply out brackets or expand our brackets, we multiply the outside number by the inside numbers so we can remove the brackets. What we are doing when we factorise is the reverse and adding those brackets back in.

All we need to do is follow these steps:

1. 1. Identify the biggest common number between all of the terms.

1. 2. For each algebraic letter, remove the highest power that you can that every number has in common.

1. 3. Using the common number and letter you found, divide each item in the expression by them and put them into brackets – keep the common number and letter outside of the brackets.

Now we know the basics, let’s look at an example.

We’ve got the following expression: \(3x^2 + 6x\)

Lets go through the steps we mentioned before:

1. 1. Identify the biggest common number – here we can see that this would be 3 as it goes into both 3 and 6.

1. 2. The highest power algebraic letter would be \(x\) as this is both in \(x\) and \(x^2\).

1. 3. Now we divide both numbers by 3 and \(x\). So \(3x^2 \div 3x = x\), and \(6x \div 3x = 2\).

So all that’s left to do is put the \(x + 2\) inside the bracket and leave the \(3x\) on the outside. Our final answer is therefore \(3x(x+2)\).

If you want to double-check you’ve factorised it right, you can go ahead and expand out the brackets and see if you get back to \(3x^2+6x\).

Difference Of The Squares

The final part of our factorising video is all about something called the ‘Difference Of The Squares’.

The key formula to remember here is as follows:

\(x^2 – y^2 = (x+y)(x-y)\)

These can crop up in all sorts of questions but it’s a ‘need-to-know’ formula.

Let’s dive into an example:

\(25x^2 – 16y^2\)

Here you can see that the 25 and 16 are both square numbers so they can be square rooted to equal 5 and 4 respectively.

Lets put this into the brackets – we square root both numbers and put them in 2 lots of brackets; one with a minus and one with a plus as shown below:

\((5x+4y)(5x-4y)\)

The reason this works is that you’ll end up multiplying the \(5x\) with itself to reproduce the \(25x^2\) and the \(4y\) with itself to get back to \(16x^2\). However, the opposite signs (being a ‘-‘ and a ‘+’) are crucial as they will produce 2 lots of \(xy\) figures when you multiply the \(x\) with the \(y\) from the opposite bracket.

Having these opposite signs means that the 2 \(xy\) figures cancel so you’re only left with the \(25x^2\) and the \(16y^2\).

That concludes our factorising video – hopefully you found this helpful.

If you have any questions, please leave them in the comments and I will respond as soon as possible.

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