Mr F’s Maths Notes

Algebra

5. Double Brackets

5. Double Brackets

You knew it was coming…Just when you have got your head around how to expand single brackets, your lovely maths teacher announces it’s time to have a go at expanding double brackets.

But the good news is that it’s no more difficult than single brackets, you don’t need to learn any new skills, and you get loads more marks for doing it!

Skills you need for success…If you know about these things, you will be fine:

• How to expand single brackets (see Algebra 2. Single Brackets)

• Rules of Algebra (see Algebra 1. Rules of Algebra)

• Rules of Negative Numbers (see Number 8. Negative Numbers)

It’s all about FOIL…Now, like with most things in maths, there are a lot of different ways of expanding double brackets, and if you are happy with your way, then just stick to it, but here is how I do it.

FOIL basically tells me the order in which I need to multiply terms, because the most common mistake people make when expanding double brackets is to miss a few terms out!

F irst

( ) ( )

1. Multiply together the first terms in each bracket – remembering to include the signs in front of them

O uter2. Multiply together the terms on the outside each bracket – remembering to include the signs in front of them

I nner3. Multiply together the terms on the inside each bracket – remembering to include the signs in front of them

L ast4. Multiply together the last terms in each bracket – remembering to include the signs in front of them

Some people call this the crab claw method!

1

2

3

4

Example 1

( 6) ( 4)a a

Until you get really comfortable, there is nothing wrong with drawing the crab claw on to remind you what to multiply!

( 6) ( 4)a a

First 2 a a a

Outer 4 4 a a Inner 6 6 a a Last 2 6 4 4 Now we write down our answers, in order, remembering if there is no sign in front of our term it’s just a disguised plus!

2 4 6 24a a a Notice that the middle terms simplify to give…

2 10 24a a

Example 2

( 10) ( 8)p p

Time for the crab claw…

First 2 p p p

Outer 8 8 pp Inner 1 0 10 p p Last 10 8 8 0

Now we write down our answers, in order, making sure we get all the signs correct!

2 8 10 80p p p

2 2 80p p

( 10) ( 8)p p

Be really careful with the NEGATIVES…

Notice that the middle terms simplify to give…

Example 3

( 9) ( 2)t t

Let’s draw our smiley face…

First 2 t t t

Outer 2 2 t t Inner 9 9 t t Last 1 2 89

2 2 9 18t t t

Carefully simplify the middle terms…

2 7 18t t

Example 4

( 7) ( 9)m m

Time for another smiley face…

First 2 m m m

Outer 9 9 mm Inner 7 7 m m Last 7 9 63

You have to know your Rules of Negative Numbers inside out for this next bit…

2 9 7 63m m m

2 16 63m m

Be so, so, so careful with the NEGATIVES…

Writing down our answers, we get…

( 9) ( 2)t t

Again, we must watch those NEGATIVES…

Once again, the signs are the key to success!

( 7) ( 9)m m

Let’s take a moment to reflect…Just before we look at a few more difficult ones (which, by the way, follow the exact same rules as these), I just want to draw your attention to the answers we got…

( 6) ( 4)a a

( 10) ( 8)p p

( 9) ( 2)t t

( 7) ( 9)m m

2 10 24a a 2 2 80p p

2 7 18t t

2 16 63m m

Now, look at the numbers in the questions and the numbers in the answers.

Can you see a quick way of getting from one to the other?...

Don’t worry if you can’t, but if you can then you are one step ahead, because that is the key to success at 6. More Factorising, which is coming up soon…

But for now, how about some tricky expanding double bracket questions?...

Example 5

(5 9) ( 3)g g

Let’s draw our smiley face…

First 25 5 g gg

Outer 5 5 3 1 gg Inner 9 9 g g Last 2 3 79

25 15 9 27g g g

Carefully simplify the middle terms…

25 6 27g g

Example 6

Are you still feeling happy?…

First 2 6 3 2 c c c

Outer 3 5 15cc Inner 4 8 2 c c Last 4 5 20

26 15 8 20c c c

26 23 20c c

NEGATIVES and Rules of Algebra again…

Writing down our answers, we get…

Again, we must watch those NEGATIVES, and we must know our Rules of Algebra!

As always, the signs are the key to success!

(5 9) ( 3)g g

(3 4) (2 5)c c

(3 4) (2 5)c c

Carefully simplify the middle terms…

Example 7

( ) ( )a b c d

Let’s draw our crab claw…

First aa c c Outer ada d Inner bb c c Last bdb d

ac ad bc bd Can we simplify the middle two (or indeed, any) of the terms?... NO because there are NO LIKE TERMS!

Example 8 – because I am feeling nasty…

Are you still smiling now?…

First 2 37 5 35a bb a a

Outer 7 4 12ab c abc Inner 2 2 13 5 5 bcc a ab Last 2 3 62bc c bc

3 2 235 b 14 15 6a abc a bc bc

Okay, you would be really unlucky to ever get one as hard as this, but there’s no reason we can’t do it

Phew! Writing down our answers, we get…

Again, we must watch those NEGATIVES, and we must know our Rules of Algebra!

As always, the signs are the key to success!

2(7 3 ) (5 2 )ab bc a c

( ) ( )a b c d 2(7 3 ) (5 2 )ab bc a c

Can we simplify the middle two (or indeed, any) of the terms?... NO because there are NO LIKE TERMS!

Last one, I promise…How would you do this one?...

2( 7)a

If you said: “well, it’s dead easy, isn’t it, the answer is just…

2 49a Then please never say that again… because it’s wrong!

Remember: squaring something means multiplying it by itself.So, this question could actually be written as…

( 7) ( 7)a a

Which means we can go back to our friend FOIL, and everyone is happy!Incidentally, if you want to check you can still do these, the final simplified answer is…

2 14 49a a

Can you see how we could have reached that answer a quicker way?…

TO BE CONTINUED on a maths website near you…

Good luck with your revision!