1. “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.

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

3. Module C4
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4. 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. 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. 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.

7. e.g.
( since is defined as 1 )

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

9. 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 )

10. 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.

11. 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.

12. 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.

13. 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.

14. 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.

15. 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.

16. 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,

17. 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.,

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

20. 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

25. 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.

26. 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.

27. 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

28. 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 )

29. 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

30. 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 )

31. 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 )

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

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

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

35. 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.

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

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

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

39. 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

40. signs
numbers
letters
There are 4 minus signs here NOT 3

41. signs
numbers
letters

42. signs
numbers
letters

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

44. signs
numbers
letters

45. signs
numbers
letters

46. 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.

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

48. (ii)
Replace n by
Valid for

49. (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

50. 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.

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

52. Solutions: 2.
Replace n by
Valid for

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

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. 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. 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. 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 )

59. 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:

60. (ii)
Replace n by
Valid for

61. (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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