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The following slides show one of the 55 presentations that cover the A2 Mathematics core modules C3 and C4. Demo Disc Teach A Level Maths Vol. 2: A2 Core.

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Presentation on theme: "The following slides show one of the 55 presentations that cover the A2 Mathematics core modules C3 and C4. Demo Disc Teach A Level Maths Vol. 2: A2 Core."— Presentation transcript:

1 The following slides show one of the 55 presentations that cover the A2 Mathematics core modules C3 and C4. Demo Disc Teach A Level Maths Vol. 2: A2 Core Modules

2 44: More Binomial Expansions © Christine Crisp Teach A Level Maths Vol. 2: A2 Core Modules

3 More Binomial Expansions Module C4 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

4 More Binomial Expansions In AS we developed a formula to expand expressions of the form as a finite series of terms provided that n was a positive integer. The formula is where e.g.

5 More Binomial Expansions We are now going to extend the work to expansions of where n is negative and/or a fraction. We will also find out how to expand expressions of the form e.g.

6 More Binomial Expansions Lets look again at where n is a positive integer Well take n = 8 as an example. Using we get Instead of using a calculator to work out these coefficients, we can simplify them.

7 More Binomial Expansions e.g. ( since is defined as 1 )

8 More Binomial Expansions ( Notice that there are the same number of factors in the numerator and denominator. ) If we replace 8 by n

9 More Binomial Expansions We can replace each coefficient of the binomial expansion in a similar way. So, becomes It can be shown that this expansion is true even when n is not a positive integer BUT with 2 properties The series becomes infinite ( We say it only converges if ) It is only valid for

10 More Binomial Expansions SUMMARY The series is infinite The series only converges to if ( also written as ) N.B. The notation cannot be used. n can be a positive or negative fraction or a negative integer. ( If n is a positive integer the series is finite. )

11 More Binomial Expansions We arent going to prove the Binomial expansion but I can illustrate it in 2 ways. Consider I simplify each term by deciding the sign first, then the constants ( by cancelling ), and the x s last.

12 More Binomial Expansions I simplify each term by deciding the sign first, then the constants ( by cancelling ), and the x s last. We arent going to prove the Binomial expansion but I can illustrate it in 2 ways. Consider

13 More Binomial Expansions I simplify each term by deciding the sign first, then the constants ( by cancelling ), and the x s last. We arent going to prove the Binomial expansion but I can illustrate it in 2 ways. Consider

14 More Binomial Expansions I simplify each term by deciding the sign first, then the constants ( by cancelling ), and the x s last. We arent going to prove the Binomial expansion but I can illustrate it in 2 ways. Consider

15 More Binomial Expansions I simplify each term by deciding the sign first, then the constants ( by cancelling ), and the x s last. We arent going to prove the Binomial expansion but I can illustrate it in 2 ways. Consider

16 More Binomial Expansions Using the binomial theorem, we have Now the l.h.s. equals, so The r.h.s. is an infinite geometric progression with a = 1 and r = 1, so,

17 More Binomial Expansions We can also draw the graphs of the l.h.s. and the r.h.s. of our expression. Drawing both graphs, well start with and, as a first approximation, the first 2 terms of the polynomial on the r.h.s.,.

18 More Binomial Expansions

19 Remember that the expansion is only valid for but even so it is not good.

20 More Binomial Expansions We will now include one more term of the polynomial at a time and draw the new graphs. As we include more terms in the polynomial the graph should get closer to the graph of So we will have and etc.

21 More Binomial Expansions

22

23

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25 Also, the graphs show very clearly that the expansion is not valid outside these values. The fit is now very good between x = 1 and x = 1 Although we havent proved the binomial expansion, we have seen that it works in this example.

26 More Binomial Expansions In AS Maths we saw how to expand binomial series when n was a positive integer. e.g. If n = 4, Our new version of the formula works for positive integers also, as we eventually reach a term that is zero. All the terms after this will also be zero, so the series is finite ( as before ). We will now apply the formula to expressions where n is not a positive integer.

27 More Binomial Expansions e.g.2 Find the 1 st 4 terms in ascending powers of x of the following. For each one, give the values of x for which the expansion is valid. (i) (ii) (iii) Solution: (i) For, replace n by

28 More Binomial Expansions Solution: (i) For, replace n by ( 2 ) (i) (ii) (iii) e.g.2 Find the 1 st 4 terms in ascending powers of x of the following. For each one, give the values of x for which the expansion is valid.

29 More Binomial Expansions Solution: (i) For, replace n by replace x by ( 2 ) (i) (ii) (iii) e.g.2 Find the 1 st 4 terms in ascending powers of x of the following. For each one, give the values of x for which the expansion is valid.

30 More Binomial Expansions Solution: (i) For, replace n by replace x by ( x ) ( 2 ) (i) (ii) (iii) e.g.2 Find the 1 st 4 terms in ascending powers of x of the following. For each one, give the values of x for which the expansion is valid.

31 More Binomial Expansions Solution: (i) For, replace n by ( 2 ) (i) (ii) (iii) e.g.2 Find the 1 st 4 terms in ascending powers of x of the following. For each one, give the values of x for which the expansion is valid. replace x by ( x )

32 More Binomial Expansions Solution: (i) The brackets are essential ! For, replace n by replace x by ( x ) ( 2 )

33 More Binomial Expansions Solution: (i) The brackets are essential ! For, replace n by replace x by ( x ) ( 2 )

34 More Binomial Expansions Solution: (i) The brackets are essential ! For, replace n by replace x by ( x ) ( 2 )

35 More Binomial Expansions Solution: (i) The brackets are essential ! I go straight to ( 3) here... For, replace n by replace x by ( x ) ( 2 ) ( since ( 2 1) spreads the expression out too much ), BUT it is the only simplification I make at this stage.

36 More Binomial Expansions Solution: (i) The brackets are essential ! For, replace n by replace x by ( x ) ( 2 )

37 More Binomial Expansions Solution: (i) The brackets are essential ! For, replace n by ( x ) ( 2 ) replace x by

38 More Binomial Expansions Solution: (i) The brackets are essential ! For, replace n by replace x by ( x ) ( 2 )

39 More Binomial Expansions As before, we simplify in the order: signs, numbers, letters Tip: Dont use a calculator to simplify the terms. Its slow, fiddly and prone to error. ( You will have to practise doing it without and be systematic ! )

40 More Binomial Expansions There are 4 minus signs here NOT 3 signsnumbers letters

41 More Binomial Expansions signsnumbers letters

42 More Binomial Expansions signsnumbers letters

43 More Binomial Expansions signsnumbers letters How many minus signs here? ANS: 6

44 More Binomial Expansions signsnumbers letters

45 More Binomial Expansions signsnumbers letters

46 More Binomial Expansions signsnumbers letters Valid for N.B. Its convenient to write the denominators as factorials. If you do this, take care with the cancelling. Ill do the next example this way.

47 More Binomial Expansions (ii) To avoid piled up fractions, we can put the factorial under the Replace n by

48 More Binomial Expansions (ii) Replace n by Valid for

49 More Binomial Expansions (iii) As weve just worked out the expansion for we can obtain the new expansion by replacing x by ( 2x ) in that answer. Valid for

50 More Binomial Expansions Exercises Write down the 1 st 4 terms, in the expansions in ascending powers of x, for the following and give the values of x for which the expansions are valid

51 More Binomial Expansions Replace n by ( 3 ) and x by ( x ) Solutions: 1. Valid for

52 More Binomial Expansions Solutions: 2. Replace n by Valid for

53 More Binomial Expansions Replace n by and x by ( 3x ) Solutions: 3. Valid for

54 More Binomial Expansions

55 The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as Handouts with up to 6 slides per sheet.

56 More Binomial Expansions In AS we developed a formula to expand expressions of the form as a finite series of terms provided that n was a positive integer. The method can be extended to expressions of the form where n is negative and/or a fraction. and

57 More Binomial Expansions SUMMARY The series is infinite The series only converges to if ( also written as ) N.B. The notation cannot be used. n can be a positive or negative fraction or a negative integer. ( If n is a positive integer the series is finite. )

58 More Binomial Expansions e.g.2 Find the 1 st 4 terms in ascending powers of x of the following. For each one, give the values of x is the expansion valid. Solution: (i) For, replace n by (i) (ii) (iii) replace x by ( x ) ( 2 )

59 More Binomial Expansions Valid for N.B. Its convenient to write the denominators as factorials. If you do this, take care with the cancelling. signs numbers letters Simplify in the order:

60 More Binomial Expansions (ii) Replace n by Valid for

61 More Binomial Expansions (iii) As weve just worked out the expansion for we can obtain the new expansion by replacing x by ( 2x ) in that answer. Valid for

62 Full version available from:- Chartwell-Yorke Ltd. 114 High Street, Belmont Village, Bolton, Lancashire, BL7 8AL England, tel (+44) (0) , fax (+44) (0)


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