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**Mr Barton’s Maths Notes**

Algebra 5. Double Brackets

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**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)

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**F irst O uter I nner L ast 1. 2. 3. 4. 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! 1 Some people call this the smiley face method! 4 3 2 F irst 1. Multiply together the first terms in each bracket – remembering to include the signs in front of them O uter 2. Multiply together the terms on the outside each bracket – remembering to include the signs in front of them I nner 3. Multiply together the terms on the inside each bracket – remembering to include the signs in front of them L ast 4. Multiply together the last terms in each bracket – remembering to include the signs in front of them

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**Example 1 Example 2 First First Outer Outer Inner Inner Last Last**

Until you get really comfortable, there is nothing wrong with drawing the smiley face on to remind you what to multiply! Time for the smiley face… Be really careful with the NEGATIVES… First First Outer Outer Inner Inner Last Last 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! Now we write down our answers, in order, making sure we get all the signs correct! Notice that the middle terms simplify to give… Notice that the middle terms simplify to give…

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**Example 3 Example 4 First First Outer Outer Inner Inner Last Last**

Let’s draw our smiley face… Time for another smiley face… Again, we must watch those NEGATIVES… Be so, so, so careful with the NEGATIVES… First First Outer Outer Inner Inner Last Last Once again, the signs are the key to success! Writing down our answers, we get… Carefully simplify the middle terms… You have to know your Rules of Negative Numbers inside out for this next bit…

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**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… 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?...

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**Example 5 Example 6 First First Outer Outer Inner Inner Last Last**

Let’s draw our smiley face… Are you still feeling happy?… Again, we must watch those NEGATIVES, and we must know our Rules of Algebra! NEGATIVES and Rules of Algebra again… First First Outer Outer Inner Inner Last Last Writing down our answers, we get… As always, the signs are the key to success! Carefully simplify the middle terms… Carefully simplify the middle terms…

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**Example 8 – because I am feeling nasty…**

Let’s draw our smiley face… Are you still smiling now?… Again, we must watch those NEGATIVES, and we must know our Rules of Algebra! Okay, you would be really unlucky to ever get one as hard as this, but there’s no reason we can’t do it First First Outer Outer Inner Inner Last Last As always, the signs are the key to success! Phew! Writing down our answers, we get… Can we simplify the middle two (or indeed, any) of the terms?... NO because there are NO LIKE TERMS! Can we simplify the middle two (or indeed, any) of the terms?... NO because there are NO LIKE TERMS!

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**TO BE CONTINUED on a maths website near you…**

Last one, I promise… How would you do this one?... If you said: “well, it’s dead easy, isn’t it, the answer is just… 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… 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… Can you see how we could have reached that answer a quicker way?… TO BE CONTINUED on a maths website near you…

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**Good luck with your revision!**

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