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![Page 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/1.jpg)
The following slides show one of the 55 presentations that cover the A2 Mathematics core modules C3
and C4.
Demo DiscDemo Disc
““Teach A Level Maths”Teach A Level Maths”Vol. 2: A2 Core ModulesVol. 2: A2 Core Modules
![Page 2: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/2.jpg)
44: More 44: More Binomial Binomial ExpansionsExpansions
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules
![Page 3: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/3.jpg)
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"
![Page 4: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/4.jpg)
More Binomial Expansions
In AS we developed a formula to expand expressions of the formnx)1( as a finite series of terms provided that n was a positive integer.The formula is
nnnnn xxCxCCx ...)1( 2210
!!
!
128
20
!!
!
)( rnr
nC r
n
wher
e
820Ce.g
.
![Page 5: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/5.jpg)
More Binomial Expansions
We are now going to extend the work to expansions of
nx)1(
where n is negative and/or a fraction.
We will also find out how to expand expressions of the form
nbxa )(
e.g. ,1)1( x ,21
)4( x 3)2( x
![Page 6: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/6.jpg)
More Binomial Expansions
Let’s look again at where n is a positive integer
nx)1(
822
81
80
88 ...)1( xxCxCCx
We’ll take n = 8 as an example.
!!
!
)( rnr
nC r
n
Using
we get
828
08
8...
62
8
71
8
80
8)1( xxxx
!!
!
!!
!
!!
!
!!
!
Instead of using a calculator to work out these coefficients, we can simplify them.
![Page 7: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/7.jpg)
More Binomial Expansions
12345123
12345678
828
08
8...
62
8
71
8
80
8)1( xxxx
!!
!
!!
!
!!
!
!!
!
e.g.!!
!
80
81
1
!!
!
71
8
12345671
12345678
!!
!
62
812345612
12345678
!!
!
53
8
( since is defined as 1 )
!0
1
8
12
78
123
678
![Page 8: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/8.jpg)
More Binomial Expansions
123
678
!5!3
!83
8
C
123
)2)(1(
)!(!3
!3
nnn
rn
nCn
( Notice that there are the same number of factors in the numerator and denominator. )
If we replace 8 by n
![Page 9: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/9.jpg)
More Binomial Expansions
We can replace each coefficient of the binomial expansion in a similar way.
So,nnnnn xxCxCCx ...)1( 2
210
becomes
32
123
)2)(1(
12
)1(1)1( x
nnnx
nnxnx n
nx...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
11 x( We say it only converges if )
• It is only valid for 11 x
![Page 10: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/10.jpg)
More Binomial ExpansionsSUMMAR
Y...
123
)2)(1(
12
)1(1)1( 32
xnnn
xnn
xnx n
• The series is infinite
• The series only converges to if
11 x
nx)1(
( also written as )1xN.B. The notation cannot be
used. rnC
• n can be a positive or negative fraction or a negative integer.( If n is a positive integer the series is
finite. )
![Page 11: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/11.jpg)
More Binomial Expansions
xx )1(1)1( 1
We aren’t going to prove the Binomial expansion but I can illustrate it in 2 ways.
1nConsider
...123
)2)(1(
12
)1(1)1( 32
xnnn
xnn
xnx n
1x
1)1( 1 x
I simplify each term by deciding the sign first, then the constants ( by cancelling ), and the x’s last.
2
12
)2)(1(x ...
123
)3)(2)(1( 3
x
![Page 12: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/12.jpg)
More Binomial Expansions
xx )1(1)1( 1
I simplify each term by deciding the sign first, then the constants ( by cancelling ), and the x’s last.
x
1)1( 1 x
We aren’t going to prove the Binomial expansion but I can illustrate it in 2 ways.
1nConsider
...123
)2)(1(
12
)1(1)1( 32
xnnn
xnn
xnx n
1x
2
12
)2)(1(x ...
123
)3)(2)(1( 3
x
![Page 13: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/13.jpg)
More Binomial Expansions
2
12
)2)(1(x
xx )1(1)1( 1
I simplify each term by deciding the sign first, then the constants ( by cancelling ), and the x’s last.
x1)1( 1 x 2x
We aren’t going to prove the Binomial expansion but I can illustrate it in 2 ways.
1nConsider
...123
)2)(1(
12
)1(1)1( 32
xnnn
xnn
xnx n
1x
...123
)3)(2)(1( 3
x
![Page 14: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/14.jpg)
More Binomial Expansions
...123
)3)(2)(1( 3
x
I simplify each term by deciding the sign first, then the constants ( by cancelling ), and the x’s last.
x1)1( 1 x 2x 3x
We aren’t going to prove the Binomial expansion but I can illustrate it in 2 ways.
1nConsider
...123
)2)(1(
12
)1(1)1( 32
xnnn
xnn
xnx n
1x
xx )1(1)1( 1
2
12
)2)(1(x
![Page 15: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/15.jpg)
More Binomial Expansions
I simplify each term by deciding the sign first, then the constants ( by cancelling ), and the x’s last.
xx )1(1)1( 1
x1)1( 1 x 2x 3x ...
We aren’t going to prove the Binomial expansion but I can illustrate it in 2 ways.
1nConsider
...123
)2)(1(
12
)1(1)1( 32
xnnn
xnn
xnx n
1x
2
12
)2)(1(x ...
123
)3)(2)(1( 3
x
![Page 16: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/16.jpg)
More Binomial Expansions
...1)1( 321 xxxx
Using the binomial theorem, we have
Now the l.h.s. equals , sox1
1
...11
1 32
xxxx
The r.h.s. is an infinite geometric progression with a = 1 and r = 1, so,
)(1
1
xS
r
aS
1
x
1
1... shl
![Page 17: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/17.jpg)
More Binomial Expansions
...11
1 32
xxxx
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
xy
1
1
xy 1polynomial on the r.h.s., .
![Page 18: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/18.jpg)
More Binomial Expansions
xy
1
1
![Page 19: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/19.jpg)
More Binomial Expansions
Remember that the expansion is only valid for
1xbut even so it is not good.
xy 1
xy
1
1
![Page 20: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/20.jpg)
More Binomial ExpansionsWe 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 x
y
1
1
So we will have x
y
1
1
21 xxy
and xy 1
321 xxxy etc.
![Page 21: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/21.jpg)
More Binomial Expansions
21 xxy
xy
1
1
![Page 22: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/22.jpg)
More Binomial Expansions
321 xxxy
xy
1
1
![Page 23: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/23.jpg)
More Binomial Expansions
4321 xxxxy
xy
1
1
![Page 24: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/24.jpg)
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54321 xxxxxy
xy
1
1
![Page 25: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/25.jpg)
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Also, the graphs show very clearly that the expansion is not valid outside these values.
115432 ...1 xxxxxxy
The fit is now very good between x = 1 and x = 1
xy
1
1
Although we haven’t proved the binomial expansion, we have seen that it works in this example.
![Page 26: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/26.jpg)
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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.
012345
))(3)(2)(1( 5
xnnnn 4n
![Page 27: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/27.jpg)
More Binomial Expansionse.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
)2)1( x 2
1
)1( x (iii) 21
)21( x
Solution: (i)
...123
)2)(1(
12
)1(1)1(
32
xnnnxnn
nxx n
For , replace n by)1( x2
![Page 28: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/28.jpg)
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Solution: (i)
...123
)2)(1(
12
)1(1)1(
32
xnnnxnn
nxx n
For , replace n by ( 2 ))1( x2
(i) (ii)
2)1( x 21
)1( x (iii) 21
)21( x
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.
![Page 29: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/29.jpg)
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Solution: (i)
...123
)2)(1(
12
)1(1)1(
32
xnnnxnn
nxx n
For , replace n by
replace x by
( 2 ))1( x2
(i) (ii)
2)1( x 21
)1( x (iii) 21
)21( x
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.
![Page 30: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/30.jpg)
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Solution: (i)
...123
)2)(1(
12
)1(1)1(
32
xnnnxnn
nxx n
For , replace n by
replace x by( x )( 2 ))1( x
2
(i) (ii)
2)1( x 21
)1( x (iii) 21
)21( x
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.
![Page 31: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/31.jpg)
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Solution: (i)
...123
)2)(1(
12
)1(1)1(
32
xnnnxnn
nxx n
For , replace n by ( 2 ))1( x2
(i) (ii)
2)1( x 21
)1( x (iii) 21
)21( x
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.
replace x by( x )
![Page 32: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/32.jpg)
More Binomial ExpansionsSolution: (i)
1)1( 2x
The brackets are essential !
...12312
1)1(32
nx )1( nnnx x )2)(1( nnn x
For , replace n by
replace x by( x )( 2 ))1( x
2
![Page 33: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/33.jpg)
More Binomial Expansions
...12312
1)1(32
nx
Solution: (i)
1)1( 2x
The brackets are essential !
)1( nn x )2)(1( nnn x
For , replace n by
replace x by( x )( 2 ))1( x
nx
)2(
2
![Page 34: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/34.jpg)
More Binomial Expansions
...12312
1)1(32
nx
Solution: (i)
1)1( 2x )2(
The brackets are essential !
)1( nnnx x )2)(1( nnn x
For , replace n by
replace x by( x )( 2 ))1( x
2
)( x
![Page 35: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/35.jpg)
More Binomial Expansions
...12312
1)1(32
nx
Solution: (i)
1)1( 2x )2(
The brackets are essential !
I go straight to (3) here . . .
)1( nnnx x )2)(1( nnn x
For , replace n by
replace x by( x )( 2 ))1( x
2
)( x )2( )3(
( since (21) spreads the expression out too much ),BUT it is the only simplification I make at this stage.
![Page 36: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/36.jpg)
More Binomial Expansions
...12312
1)1(32
nx
Solution: (i)
1)1( 2x )2(
The brackets are essential !
)1( nnnx x )2)(1( nnn x
For , replace n by
replace x by( x )( 2 ))1( x
2
)( x )2(12)3( )( x 2
![Page 37: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/37.jpg)
More Binomial Expansions
...12312
1)1(32
nx
Solution: (i)
1)1( 2x )2(
The brackets are essential !
)1( nnnx x )2)(1( nnn x
For , replace n by
( x )( 2 ))1( x
2
)( x )2(12)3( )( x 2
)2( )3(
)4(
replace x by
![Page 38: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/38.jpg)
More Binomial Expansions
...12312
1)1(32
nx
Solution: (i)
1)1( 2x )2(
The brackets are essential !
)1( nnnx x )2)(1( nnn x
For , replace n by
replace x by( x )( 2 ))1( x
2
)( x )2(12)3( )( x 2
)2(123
)3( )( x 3
)4(
![Page 39: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/39.jpg)
More Binomial Expansions
As before, we simplify in the order:
signs, numbers, letters
1)1( 2x )2( )( x )2(12)3( )( x 2
)2(123
)3( )( x 3
)4(
x21 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 ! )
![Page 40: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/40.jpg)
More Binomial Expansions
1)1( 2x )2( )( x )2(12)3( )( x 2
)2(123
)3( )( x 3
)4(
x21
There are 4 minus signs here NOT 3
signsnumber
sletters
![Page 41: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/41.jpg)
More Binomial Expansions
1)1( 2x )2( )( x )2(12)3( )( x 2
)2(123
)3( )( x 3
)4(
x21
signsnumber
sletters
3
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1)1( 2x )2( )( x )2(12)3( )( x 2
)2(123
)3( )( x 3
)4(
x21
signsnumber
sletters
2x3
![Page 43: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/43.jpg)
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1)1( 2x )2( )( x )2(12)3( )( x 2
)2(123
)3( )( x 3
)4(
x21
signsnumber
sletters
2x How many minus signs here?
ANS: 6
3
![Page 44: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/44.jpg)
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1)1( 2x )2( )( x )2(12)3( )( x 2
)2(123
)3( )( x 3
)4(
x21
signsnumber
sletters
2x 43
![Page 45: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/45.jpg)
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1)1( 2x )2( )( x )2(12)3( )( x 2
)2(123
)3( )( x 3
)4(
x21
signsnumber
sletters
2x 4 3x3
![Page 46: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/46.jpg)
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1)1( 2x )2( )( x )2(12)3( )( x 2
)2(123
)3( )( x 3
)4(
x21
signsnumber
sletters
2x 4 3x3
Valid for 1xN.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.
![Page 47: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/47.jpg)
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(ii)
21
)1( x
...12312
1)1(32
nx )1( nnnx x )2)(1( nnn x
To avoid piled up fractions, we can put the factorial under the
2x
Replace n by2
1
1)1( 21
x
2
1x
2
1
2
1
!2
2x
![Page 48: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/48.jpg)
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(ii)
21
)1( x
...12312
1)1(32
nx )1( nnnx x )2)(1( nnn x
1)1( 21
x
2
1x
2
1
2
1
!2
2x
2
1
2
1
2
3
!3
3x
1 8
2x 16
23x
2
x
Replace n by2
1
Valid for 1x
![Page 49: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/49.jpg)
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16821)1(
3221 xxx
x
(iii) 21
)21( xAs we’ve just worked out the expansion for
21
)1( xwe can obtain the new expansion by replacing x by ( 2x ) in that answer.
1)21( 21
x2
)2( x8
)2( 2x
16
)2( 3x
1 x 8
4 2x
2
16
8 3x
2
221
32 xxx Valid for 12 x 2
1 x
![Page 50: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/50.jpg)
More Binomial ExpansionsExercis
esWrite 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. 3)1( x 41
)1( x 31
)31(
x2. 3.
![Page 51: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/51.jpg)
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...12312
1)1(32
nx )1( nnnx x )2)(1( nnn x
Replace n by ( 3 ) and x by ( x )
1)1( 3x
!2
))(4)(3( 2x
!3
))(5)(4)(3( 3x
x31 26x 310x
2
2
2
3)1( xSolutions: 1.
))(3( x
Valid for 1x
![Page 52: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/52.jpg)
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...12312
1)1(32
nx )1( nnnx x )2)(1( nnn x
1)1( 41
x!24
3
4
1 2x
!34
7
4
3
4
1 3x
41
x
32
3 2x
128
7 3x
41
)1( xSolutions: 2.
x
4
1
Replace n by )( 41
2
Valid for 1x
![Page 53: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/53.jpg)
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...12312
1)1(32
nx )1( nnnx x )2)(1( nnn x
1)31( 31
x!2
)3(
3
4
3
1 2x
!3
)3(
3
7
3
4
3
1 3x
x1 22x3
14 3x
)3(3
1x
Replace n by and x by ( 3x )
)( 31
3
Solutions: 3. 31
)31(
x
2
2
31 x
Valid for 13 x
![Page 54: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/54.jpg)
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![Page 55: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/55.jpg)
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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.
![Page 56: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/56.jpg)
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In AS we developed a formula to expand expressions of the formnx)1( as a finite series of terms provided that n was a positive integer.
The method can be extended to expressions of the form
nx)1( where n is negative and/or a fraction.
nbxa )( and
![Page 57: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/57.jpg)
More Binomial ExpansionsSUMMAR
Y...
123
)2)(1(
12
)1(1)1( 32
xnnn
xnn
xnx n
• The series is infinite
• The series only converges to if
11 x
nx)1(
( also written as )1xN.B. The notation cannot be
used. rnC
• n can be a positive or negative fraction or a negative integer.( If n is a positive integer the series is
finite. )
![Page 58: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/58.jpg)
More Binomial Expansionse.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)
...123
)2)(1(
12
)1(1)1(
32
xnnnxnn
nxx n
For , replace n by)1( x2
(i) (ii)
2)1( x 21
)1( x (iii) 21
)21( x
replace x by( x )( 2 )
1)1( 2x )2( )( x )2(12)3( )( x 2
)2(123
)3( )( x 3
)4(
![Page 59: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/59.jpg)
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1)1( 2x )2( )( x )2(12)3( )( x 2
)2(123
)3( )( x 3
)4(
x21 2x 4 3x3
Valid for 1xN.B. It’s convenient to write the denominators
as factorials. If you do this, take care with the cancelling.
signsnumber
sletters
Simplify in the order:
![Page 60: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/60.jpg)
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(ii)
21
)1( x
1)1( 21
x
2
1x
2
1
2
1
!2
2x
2
1
2
1
2
3
!3
3x
1 8
2x 16
23x
2
x
Replace n by2
1
Valid for 1x
...123
)2)(1(
12
)1(1)1(
32
xnnnxnn
nxx n
![Page 61: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/61.jpg)
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16821)1(
2221 xxx
x
(iii) 21
)21( xAs we’ve just worked out the expansion for
21
)1( xwe can obtain the new expansion by replacing x by ( 2x ) in that answer.
1)21( 21
x2
)2( x8
)2( 2x
16
)2( 3x
1 x 8
4 2x
2
16
8 3x
2
221
32 xxx Valid for 12 x 2
1 x
![Page 62: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070305/551464cc550346284e8b5aaf/html5/thumbnails/62.jpg)
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