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! 2 Some people call this the crab claw method! 1 ( ) ( 3 ) 4 1. F irst Multiply together the first terms in each bracket – remembering to include the signs in front of them 2. O uter Multiply together the terms on the outside each bracket – remembering to include the signs in front of them 3. I nner Multiply together the terms on the inside each bracket – remembering to include the signs in front of them 4. L ast Multiply together the last terms in each bracket – remembering to include the signs in front of them Example 1 Example 2 (a 6) (a 4) Until you get really comfortable, there is nothing wrong with drawing the crab claw on to remind you what to multiply! ( p 10) ( p 8) Time for the crab claw… ( p 10) ( p 8) (a 6) (a 4) Be really careful with the NEGATIVES… a a a 4 6 a 6 4 First Outer Inner Last a2 4a 6a 24 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! a 4a 6a 24 2 Notice that the middle terms simplify to give… a2 10a 24 First Outer Inner Last p p p2 p 8 8p 10 p 10 p 10 8 80 Now we write down our answers, in order, making sure we get all the signs correct! p2 8 p 10 p 80 Notice that the middle terms simplify to give… p2 2 p 80 Example 3 Example 4 (t 9) (t 2) Let’s draw our smiley face… (m 7) (m 9) Time for another smiley face… (m 7) (m 9) (t 9) (t 2) Again, we must watch those NEGATIVES… First Outer Inner Last t t t2 t 2 2t 9 t 9t 9 2 18 Once again, the signs are the key to success! t 2 2t 9t 18 Carefully simplify the middle terms… t 2 7t 18 Be so, so, so careful with the NEGATIVES… First Outer Inner Last m m m2 m 9 9m 7 m 7 m 7 9 63 Writing down our answers, we get… m2 9m 7m 63 You have to know your Rules of Negative Numbers inside out for this next bit… m2 16m 63 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… (a 6) (a 4) a2 10a 24 ( p 10) ( p 8) p2 2 p 80 (t 9) (t 2) t 2 7t 18 (m 7) (m 9) m2 16m 63 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 Example 6 (5g 9) ( g 3) Let’s draw our smiley face… (3c 4) (2c 5) Are you still feeling happy?… (5g 9) ( g 3) Again, we must watch those NEGATIVES, and we must know our Rules of Algebra! First 5g 5g 9 9 Outer Inner Last g 3 g 3 2 5g 15g 9g 27 As always, the signs are the key to success! 5g 15g 9 g 27 2 Carefully simplify the middle terms… 5g 2 6 g 27 (3c 4) (2c 5) NEGATIVES and Rules of Algebra again… First Outer Inner Last 3c 2c 6c 2 3c 5 15c 4 2c 8c 4 5 20 Writing down our answers, we get… 6c2 15c 8c 20 Carefully simplify the middle terms… 6c 2 23c 20 Example 7 (a b) (c d ) Let’s draw our crab claw… (a b) (c d ) Again, we must watch those NEGATIVES, and we must know our Rules of Algebra! First Outer Inner Last a c ac a d ad b c bc b d bd As always, the signs are the key to success! 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… (7ab 3bc) (5a2 2c) Are you still smiling now?… (7ab 3bc) (5a2 2c) 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 Outer Inner Last 7ab 5a 2 35a3b 7ab 2c 14abc 3bc 5a2 15a2bc 3bc 2c 6bc2 Phew! Writing down our answers, we get… 35a3b 14abc 15a2bc 6bc2 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?... (a 7)2 If you said: “well, it’s dead easy, isn’t it, the answer is just… a2 49 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… (a 7) (a 7) 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… a 2 14a 49 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!