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**“Teach A Level Maths” Vol. 2: A2 Core Modules**

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

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**“Teach A Level Maths” Vol. 2: A2 Core Modules**

44: More Binomial Expansions © Christine Crisp

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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"

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**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.

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**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.

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**Let’s look again at where n is a positive integer**

We’ll take n = 8 as an example. Using we get Instead of using a calculator to work out these coefficients, we can simplify them.

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e.g. ( since is defined as 1 )

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

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**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 It is only valid for ( We say it only converges if )

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

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**We aren’t 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.

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**We aren’t 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.

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**We aren’t 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.

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**We aren’t 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.

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**We aren’t 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.

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**a = 1 and r = -1, so, 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,

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**We can also draw the graphs of the l. h. s. and the r. h. s**

We can also draw the graphs of the l.h.s. and the r.h.s. of our expression. Drawing both graphs, we’ll start with and, as a first approximation, the first 2 terms of the polynomial on the r.h.s.,

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**Remember that the expansion is only valid for**

but even so it is not good.

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**We will now include one more term of the polynomial at a time and draw the new graphs.**

So we will have and etc. As we include more terms in the polynomial the graph should get closer to the graph of

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**The fit is now very good between x = -1 and x = 1**

Also, the graphs show very clearly that the expansion is not valid outside these values. Although we haven’t proved the binomial expansion, we have seen that it works in this example.

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**In AS Maths we saw how to expand binomial series when n was a positive integer.**

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 ). e.g. If n = 4, We will now apply the formula to expressions where n is not a positive integer.

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**e. g. 2 Find the 1st 4 terms in ascending powers of x of the following**

e.g.2 Find the 1st 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

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**e. g. 2 Find the 1st 4 terms in ascending powers of x of the following**

e.g.2 Find the 1st 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 ( -2 )

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**e. g. 2 Find the 1st 4 terms in ascending powers of x of the following**

e.g.2 Find the 1st 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 ( -2 ) replace x by

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**e. g. 2 Find the 1st 4 terms in ascending powers of x of the following**

e.g.2 Find the 1st 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 ( -2 ) replace x by ( -x )

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**e. g. 2 Find the 1st 4 terms in ascending powers of x of the following**

e.g.2 Find the 1st 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 ( -2 ) replace x by ( -x )

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Solution: (i) For , replace n by ( -2 ) replace x by ( - x ) The brackets are essential !

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Solution: (i) For , replace n by ( -2 ) replace x by ( - x ) The brackets are essential !

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Solution: (i) For , replace n by ( -2 ) replace x by ( - x ) The brackets are essential !

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

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Solution: (i) For , replace n by ( -2 ) replace x by ( - x ) The brackets are essential !

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Solution: (i) For , replace n by ( -2 ) replace x by ( - x ) The brackets are essential !

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Solution: (i) For , replace n by ( -2 ) replace x by ( - x ) The brackets are essential !

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**Tip: Don’t use a calculator to simplify the terms**

Tip: Don’t use a calculator to simplify the terms. It’s slow, fiddly and prone to error. ( You will have to practise doing it without and be systematic ! ) As before, we simplify in the order: signs, numbers, letters

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signs numbers letters There are 4 minus signs here NOT 3

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signs numbers letters

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signs numbers letters

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signs numbers letters How many minus signs here? ANS: 6

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signs numbers letters

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signs numbers letters

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signs numbers letters Valid for N.B. It’s convenient to write the denominators as factorials. If you do this, take care with the cancelling. I’ll do the next example this way.

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

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(ii) Replace n by Valid for

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(iii) As we’ve just worked out the expansion for we can obtain the new expansion by replacing x by ( -2x ) in that answer. Valid for

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Exercises Write down the 1st 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. 1. 2. 3.

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Solutions: 1. Replace n by ( -3 ) and x by ( -x ) Valid for

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Solutions: 2. Replace n by Valid for

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Solutions: 3. Replace n by and x by ( 3x ) Valid for

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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.

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**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

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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. )

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**e. g. 2 Find the 1st 4 terms in ascending powers of x of the following**

e.g.2 Find the 1st 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 )

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Valid for N.B. It’s convenient to write the denominators as factorials. If you do this, take care with the cancelling. signs numbers letters Simplify in the order:

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(ii) Replace n by Valid for

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(iii) As we’ve just worked out the expansion for we can obtain the new expansion by replacing x by ( -2x ) in that answer. Valid for

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