Expanding Double/Triple Brackets - Go Nuts
Nuts. The ultimate brain food…
If you want any hope in passing your gcse maths exam and if you want a half decent grade, you need to master expanding brackets (and factorising, but we’ll do that in another post.)
Expanding two (or more) brackets is where you need to be super careful.
Minus numbers can really trip you up.
However if you do it step by step, its fairly straightforward.
Double brackets are similar to expand, the hard part is making sure you multiply everything, without missing a pair.
E.g. if you have (x + 2)(x + 3)
Multiply everything in the second bracket by x, then by 2 and then simplify.
How to multiply double brackets
So the simplified answer is: x^2 + 6x + 6
There are a couple of methods that help you to make sure you don’t miss anything.
The FOIL method. Not my favourite, but worth a mention.
FOIL stands for First, Outside, Inside, Last.
Its a sequence to follow when multiplying out the numbers.
So with the example above you would do: First, meaning the first number in each bracket: x times x
Then the outside numbers: x times 3
Then the numbers that are both on the inside: 2 times x
Finally, the numbers that are last in each bracket: 2 times 3
If you understand that, brilliant! If you can’t get your head around it, that’s fine. There is another way:
2. The Parrot’s Claw
If you look at the diagram above you can (hopefully) see the arrows I’ve drawn make the shape of a parrots claw. Or it might look like a crab’s claw, but either way its some kind of claw shape.
The point is, it doesn’t really matter. You just need a method or some way to check that you have multiplied everything.
Another way to check, if you are multiplying out double brackets, where the letter is the same. i.e. x is in both, you will end up with a quadratic. (we will cover quadratics in more detail soon.)
How to cope with minus numbers
Expanding brackets is easy…until its not.
That’s usually when they throw in minus numbers.
So let’s use the previous example, but with a minus 3 instead:
(x + 2)(x - 3)
The first bit doesn’t change, its still x squared: x^2.
Then we have 2 times x which is 2x.
Then we do x times -3 (minus or negative 3) which is -3x.
Then we do 2 times -3 which is -6.
So our expanded expression looks like this: x^2 + 2x - 3x - 6
To tidy that up: 2x subtract or minus 3x which is -x.
So the final version is: x^2 - x -6.
The minus means you have to subtract that number or term.
Maths Practice Questions: Expanding Brackets
Practice Expanding Double Brackets
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So what about Triple Brackets?
Triple brackets are like this: (x + 2)(x + 3)(x + 4)
Which might look scary at first, but there is an easy way to do them.
Choose 2 of the brackets and multiply them out as you would any set of double brackets.
Then take the result of that and multiply all the terms by the first thing in the third bracket.
Then multiply all the terms by the second thing in the 3rd bracket.
Let’s work through that example:
(x + 2)(x + 3)
This gives us: x2 + 5x + 6
So now we need to expand out: (x + 4)(x2 + 5x + 6)Lets do everything multiplied by x: (x3 + 5x2 + 6x)
Then multiply everything by the 4: (4x2 + 20x + 24)
So we have: x3 + 5x2 + 6x + 4x2 + 20x + 24
We need to collect like terms:
x3 + 9x2 + 26x + 24
And that’s it, that’s the final answer.
So multiplying out triple brackets is not necessarily harder maths, its a pain because you have to repeat several steps.
If you do it carefully, one bit at a time, you should be fine.
Free Quiz: Expanding Triple Brackets
1. Expand: (x + 2)(x - 1)(x + 3)
