capter algebra

8/8/2019 Capter Algebra

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Chapter 5

BINOMIAL THEOREM

Enter


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WELCOMEPlease select any Topic

Combination

For n Positive

Integer

For Rational

Number

EXIT

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Definition:The number of selections or combinationschosen from a given number where order is

irrelevant.

BM

BM

nr0whereC rn

Selecting r from n number or object

is given by

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Takrif:

Bilangan pemilihan atau gabungan yangdipilih dari suatu nombor dengan

susunan yang tidak diambilkira.

BMEnglish

Pemilihan r dari sebanyak n nombor

atau objek diberikan oleh

nr0denganC rn

sebelumnya seterusnya

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BMBM

Factorial n is given by

n!=n.(n-1).(n-2).3.2.1

Thus

!r

)1rn)....(2n).(1n.(n

!r)!rn(

!n

C rn

+

=

=

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BMEnglish

n faktorial diberi oleh

n!=n.(n-1).(n-2).3.2.1

Maka

!r

)1rn)....(2n).(1n.(n

!r)!rn(

!n

C rn

+

=

=


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BMBM

Observe that

rnn

rn CC =

2nn

2n

1nn

1n

n

n

0

n

CC

CC

CC

=

=

=So we have

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BMEnglish

Perhatikan bahawa

rnn

rn CC =

2nn2n

1nn

1n

n

n

0

n

CC

CC

CC

=

=

=Oleh itu


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BMBM

If n is a positive integer and a, xthen

nx....rxrnarn

Cn....2x2na2

Cnx1na1

Cnnan)xa( ++

++++=+

next

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BMBM

Thus

( )

.x...xa!r

)1rn)..(1n(n...

.....xa!2

)1n(nxnaaxa

nrrn

22n1nnn

++++

+++=+

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Jika n ialah integer positif dan a, xmaka

nx....rxrnarn

Cn....2x2na2

Cnx1na1

Cnnan)xa( ++++++=+

.x...xa!r

)1rn)..(1n(n...

.....xa!2

)1n(nxnaa

nrrn

22n1nn

+++

+

+

++=

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BMBM

The expansion above has thefollowing characters

(a +x) has (n + 1) termsPower summation for every a and x

terms is n.General term is the r+1th term that is

rxrnarCn1ra

=+

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BMEnglish

Dari kembangan di atas didapatisifat-sifat berikut

(a +x)mempunyai (n + 1) sebutan

hasiltambah kuasa bagi setiapsebutan a dan x ialah n

sebutan am ialah sebutan ke r+1

iaitu rxrnarCn1ra =+


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BMBM

.....x!r

)1r)..(1(.....x

!2

)1(x1)x1( r2 +

++

++=+

Let n = , where maka

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BMEnglish

.....x!r

)1r)..(1(.....x

!2

)1(x1)x1( r2 +

++

++=+

Katakan n= , dengan maka


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BMBM

This binomial expansion

Is an indefinite series

Series exist for that is 1


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Expand (2 5x)-2 as far as the term inx3 . For what range of values of x willthe expansion be valid?

Solution

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Solution:

22

22

2

x512

2

x512)x52(

=

=

First convert to the standard form:

Now expand the binomial

++++=

+

++=

...x2

375x

4

75x512

...)2

x5

(2

)4)(3)(2(

)2

x5

(2

)3)(2(

)2

x5

)(2(12)x52(

322

3222

Return

5

2x

5

2.e.i1

2

x51


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BMEnglish

Kembangan binomial ini adalah

adalah siri tak terhingga.

siri wujud untuk iaitu 1


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BMBM

The binomial theorem may be used forsolving problem in approximation

Example 1

Example 2

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Find the value of (1.01)10 correct to

four significant figures.Solution:

.figurestsignificanfourto1046.11046221.10000021.000012.00045.01.01

...)01.0(210)01.0(120)01.0(45)01.0(101)01.01(01.0xLet

.....x210x120x45x101)x1(

43210

43210

++++

++++++

=

+++++=+

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Expand (1 - x)1/2 as far as the term in x3.

By substituting in the expansion, find

an approximation of , correct to four

significant figures.

501

2

Solution:

...16

x

8

x

2

x1)x1(

3221

=

This expansion will be valid if x is numerically

less than 1, which is true if .50

1x =

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Now let x = 1/50, then

4142.1)9899495.0(7

102

9899495.0210

7

0000005.000005.001.01

50

49

...

)50(16

1

)50(8

1

)50(2

11

50

11

32

2

1

=

=

+=

Return


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BMEnglish

Teorem Binomial boleh digunakan bagimenyelesaikan masalah penghampiran

Contoh 1

Contoh 2

seterusnya

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End of Chapter 5

capter algebra

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