29 the Binomial Expansion

8/23/2019 29 the Binomial Expansion

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The Binomial Expansion


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Powers of a + b

In this presentation we will develop a formula toenable us to find the terms of the expansion of

nba )(

where n is any positive integer.

We call the expansion binomial as the originalexpression has 2 parts.
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Powers of a + b

222 baba

2)( ba ))(( baba We know that

so the coefficients of the terms are 1, 2 and 1

We can write this as22baba 1 2 1


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2ab1ba

2 2)2)((

22bababa

Powers of a + b

3)( ba 2))(( baba 3

a1


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2ab2 3b1

223abbaa 1 2 1

)2)((22bababa

Powers of a + b

3)( ba 2))(( baba

ba21


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223abbaa 1 2 1

)2)((22bababa

Powers of a + b

3)( ba 2))(( baba

322 babba 1 2 13223babbaa 331 1


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223abbaa

Powers of a + b

3)( ba 2))(( baba )2)((

22bababa

322 babba 3223babbaa

so the coefficients of the expansion of

are 1, 3, 3 and 1

3)( ba

1 2 1

1 2 1

331 1


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Powers of a + b

4)( ba 3))(( baba )33)((

3223babbaaba 32234

abbabaa 1 3 3 143223 babbaba 1 3 3 1432234babbabaa 641 4 1
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32234abbabaa

Powers of a + b

4)( ba 3))(( baba )33)((

3223babbaaba

43223 babbaba 432234babbabaa

1 3 3

1 3 3

641 4

1

1

1

This coefficient . . .. . . is found by adding 3 and 1; the

coefficients that are in 3)( ba


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3

1

4

32234abbabaa

Powers of a + b

4)( ba 3))(( baba )33)((

3223babbaaba

43223 babbaba 432234babbabaa

1 3

3 3

61 4

1

1

1

This coefficient . . .. . . is found by adding 3 and 1; the

coefficients that are in 3)( ba


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Powers of a + b

So, we now have

3)( ba2)( ba

CoefficientsExpression

1 2 1

1 3 3 1

4)( ba 1 4 6 4 1


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So, we now have

3)( ba2)( ba


1 2 1

1 3 3 1

4)( ba 1 4 6 4 1Each number in a row can be found by adding the 2coefficients above it.

Powers of a + b


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Powers of a + b

So, we now have

3)( ba2)( ba


1 2 1

1 3 3 1

4)( ba 1 4 6 4 1The 1st and last numbers are always 1.

Each number in a row can be found by adding the 2coefficients above it.


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Powers of a + b

So, we now have

3)( ba2)( ba


1 2 1

1 3 3 1

1

)(ba

1 1

0)( ba

4)( ba 1 4 6 4 1To make a triangle of coefficients, we can fill inthe obvious ones at the top.

1


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Powers of a + b

The triangle of binomial coefficients is calledPascals triangle, after the French mathematician.

. . . but its easy to know which row we want as,for example,

3)( ba starts with 1 3 . . .10)( ba will start 1 10 . . .

Notice that the 4th row gives the coefficients of

)( ba 3


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Exercise

Find the coefficients in the expansion of6

)( baSolution: We need 7 rows

1 2 1

1 3 3 1

1 1

1

1 4 6 4 11 5 10 110 5

1 6 15 120 15 6Coefficients


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We usually want to know the complete expansion notjust the coefficients.

Powers of a + b

5)( bae.g. Find the expansion of

Pascals triangle gives the coefficientsSolution:1 5 10 110 5

The full expansion is

Tip: The powers in each term sum to 5

54322345babbababaa 1 5 10 10 5 11


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e.g. 2 Write out the expansion of inascending powers of x.

4

)1( x

Powers of a + b

Solution: The coefficients are

a 4322344 464)( a a a ab b b b bTo get we need to replace a by 14)1( x

( Ascending powers just means that the 1st termmust have the lowest power of x and then the

powers must increase. )

1 4 6 14

We know that
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14322344

464)( 1 (1) (1) (1)b b b b b


1 4 6 14

We know that

Powers of a + b


To get we need to replace aby 14)1( x

4

)1( x


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4322344464)(

Be careful! The minus sign . . .is squared as well as the x.

The brackets are vital, otherwise the signs will be wrong!


1 4 6 14

We know that

Powers of a + b


To get we need to replace aby 1 and

b by (- x)

4)1( x

1 (1)1 (1) (1)(-x) (-x) (-x) (-x) (-x)

Simplifying gives

4)1( x 1 x4 26x 34x 4x

4

)1( x


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4)1( xTo get we need to replace a by 1 and

b by (- x)

Since we know that any power of 1 equals 1, we

could have written 1 here . . .


1 4 6 14

We know that

Powers of a + b


4322344464)( 1 1 (1) (1) (1)(-x) (-x) (-x) (-x) (-x)

Simplifying gives

4)1( x 1 x4 26x 34x 4x

4

)1( x


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b by (- x)


1 4 6 14

We know that

Powers of a + b


432234464)( 1 1 (1) (1) (1)(-x) (-x) (-x) (-x) (-x)

Simplifying gives

4)1( x 1 x4 26x 34x 4x

Since we know that any power of 1 equals 1, we

could have written 1 here . . .

4

)1( x


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b by (- x)


1 4 6 14

We know that

Powers of a + b


432234464)( 1 1 (1) (1) (1)(-x) (-x) (-x) (-x) (-x)

Simplifying gives

4)1( x 1 x4 26x 34x 4x. . . and missed these 1s out.

4

)1( x


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1 4 6 14

We could go straight to

Powers of a + b


4324464)( 1 1(-x) (-x) (-x) (-x) (-x)

Simplifying gives

4)1( x 1 x4 26x 34x 4x

4

)1( x


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Exercise

1. Find the expansion of in ascendingpowers of x.

5

)21( x


1 5 10 110 5

5432)2()2(5)2(10)2(10)2(51 xxxxx

543232808040101 xxxxx

So, 5)21( x
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Powers of a + b

20)( baIf we want the first few terms of the expansionof, for example, , Pascals triangle is nothelpful.

We will now develop a method of getting the

coefficients without needing the triangle.


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Each coefficient can be found by multiplying theprevious one by a fraction. The fractions form aneasy sequence to spot.

Powers of a + b6

)(baLets considerWe know from Pascals triangle that the coefficients

are

1 6 15 115 620

1

6

2

53

44

35

26

1There is a pattern here:

So if we want the 4th coefficient without finding

the others, we would need

3

4

2

5

1

6 ( 3 fractions )


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Powers of a + b

87654321

1314151617181920

The 9th coefficient of is20)( ba

For we get20)( ba1 20 190 1140

2

193

18etc.

Even using a calculator, this is tedious to simplify.However, there is a shorthand notation that isavailable as a function on the calculator.

1

20
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87654321

1314...181920

123...12

123...12

Powers of a + b

123...181920 We write 20!

is called 20 factorial.

( 20 followed by an exclamation mark )

We can write87654321

1314151617181920

!!

!

128

20The 9th term of is20)( ba 812

128

20ba

!!

!
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Powers of a + b

!!!

12820 can also be written as 820C or

820

This notation. . .. . . gives the number of ways that 8 items canbe chosenfrom 20.

is read as 20 C 8 or 20 choose 8 andcan be evaluated on our calculators.

820C

The 9th term of is then20)( ba 812820 baC

In the expansion, we are choosing the letter b 8

times from the 20 sets of brackets that make up. ( a is chosen 12 times ).20)( ba


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Powers of a + b

The binomial expansion of is

20

)(ba2020)( aba 218220 baC

20317

3

20... bbaC

We know from Pascals triangle that the 1st twocoefficients are 1 and 20, but, to complete thepattern, we can write these using the C notation:

0

20

1 C and 120

20 C

ba1920

Since we must define 0! as

equal to 1.

1!20!0

!200

20 C

h l E


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Powers of a + b

!!

!

!!

!

812

20

128

20

Tip: When finding binomial expansions, it can be usefulto notice the following:

820CSo, is equal to 12

20C

Any term of can then be written asrr

r baC2020

20

)( bawhere r is any integer from 0 to 20.

Th B l E


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The expansion of isnx)1(

Any term of can be written in the formn

ba )( where r is any integer from 0 to n.

rrnr

nbaC

Generalizations

The binomial expansion of in ascendingpowers of x is given by

n

ba )( nnnnnnn

n

bbaCbaCaC

ba

...

)(

222

110

nnnnnxxCxCCx ...)1( 2210

Th Bi i l E i
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e.g.3 Find the first 4 terms in the expansion ofin ascending powers of x.18)1( x

2218

)( xC

Powers of a + b

Solution:

18)1( x 018C )(118 xC...)(

33

18 xC1 x18

2153 x ...816

3 x

Th Bi i l E i


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e.g.4 Find the 5th

term of the expansion ofin ascending powers of x.

12

)2( x

484 )2(

12xC

Solution: The 5th term contains 4x

Powers of a + b

It is

48)2(495 x

4126720x

These numberswill always bethe same.

Th Bi i l E i


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nba )( nnnnnnn

n

bbaCbaCaC

ba

...

)(

222

110

SUMMARY

The ( r +1 ) th term is rrnr

nbaC

The expansion of isnx)1(

nnnnnxxCxCCx ...)1( 2210

Th Bi i l E i


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Exercise

1. Find the 1st

4 terms of the expansion ofin ascending powers of x.

8

)32( x

Solution:

353

8262

871

880

8)3(2)3(2)3(22 xCxCxCC

2. Find the 6th term of the expansion ofin ascending powers of x.

13)1( x

3248384161283072256 xxx

55

13)( xC Solution:5

1287x

Th Bi i l E i


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Th Bi i l E i


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The following slides contain repeats ofinformation on earlier slides, shown withoutcolour, so that they can be printed and

photocopied.

For most purposes the slides can be printedas Handouts with up to 6 slides per sheet.


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Powers of a + b

Pascals Triangle

3)( ba2)( ba


1 2 1

1 3 3 1

1

)( ba

1 1

0)( ba

4)( ba 1 4 6 4 1

1


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We usually want to know the complete expansion notjust the coefficients.

Powers of a + b

5)( bae.g. Find the expansion of

Pascals triangle gives the coefficientsSolution:

1 5 10 110 5

The full expansion is

Tip: The powers in each term sum to 5

54322345babbababaa 1 5 10 10 5 11

l E


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1 4 6 14

So,

Powers of a + b


4324464)( 1 1(-x) (-x) (-x) (-x) (-x)

Simplifying gives

4)1( x 1 x4 26x 34x 4x

4

)1(x

h B l E


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Powers of a + b

87654321

1314151617181920

The 9th coefficient of is20)( ba

For we get20)( ba

1 20 190 1140

2

193

18etc.

Even using a calculator, this is tedious to simplify.However, there is a shorthand notation that isavailable as a function on the calculator.

1

20

Th Bi i l E i


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123...12

123...12

87654321

1314...181920

Powers of a + b

123...181920 We write 20!

is called 20 factorial.

( 20 followed by an exclamation mark )

We can write

87654321

1314151617181920

!!

!

128

20The 9th term of is20)( ba 812

128

20ba

!!

!

Th Bi i l E i


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Powers of a + b

!!

!

12820 can also be written as 820C or

8

20

This notation. . .. . . gives the number of ways that 8 items canbe chosen from 20.

is read as 20 C 8 or 20 choose 8 andcan be evaluated on our calculators.

820C

The 9th term of is then20)( ba 812820 baC

In the expansion, we are choosing the letter b 8

times from the 20 sets of brackets that make up.20)( ba

Th Bi i l E i


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Powers of a + b

!!

!

!!

!

812

20

128

20

Tip: When finding binomial expansions, it can be usefulto notice the following:

820CSo, is equal to 12

20C

Any term of can then be written asrr

r baC2020

20

)( bawhere r is any integer from 0 to 20.

Th Bi i l E i


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e.g.3 Find the first 4 terms in the expansion ofin ascending powers of x.18)1( x

2218 )( xC

Powers of a + b

Solution:

18)1( x 018C )(118 xC...)(

33

18 xC1

x18

2

153x

...816

3 x

Th Bi i l E i


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nba )(

nnnnnnn

n

bbaCbaCaC

ba

...)(

222110

SUMMARY

The ( r +1 ) th term is rrnr

nbaC

The expansion of is

nx)1(

nnnnn

xxCxCCx ...)1( 2210

29 the Binomial Expansion

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