ba 501 chapter 1 binomial expansions.. sem 5 notes

ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501

1

CHAP 1 - Binomial Expansions

The binomial theorem describes the algebraic expansion of powers of a binomial.

Figure 1 : Example use the binomial Expansion in geometric

download@

math2ever.blogspot.com

http://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Binomialhttp://math2ever.blogspot.com


2

There are 3 methods to expand binomial expression Method 1 - Algebra method Expansion two or more expression.

Example: The expansion depend on power value (n) n = 0, (a + x)0 = 1 n = 1, (a + x)1 = a + x n = 2, (a + x)2 = (a + x) (a + x) = a2 + 2ax + x2 n = 3, (a + x)3 = (a + x) (a + x) (a + x) = a3 + 3a2x + 3ax2 + x3 n = 4, (a + x)4 = (a + x)(a + x)(a + x)(a + x) = a4 +4a3x +6a2x2 +4ax3+ x4

Method 2 - PASCAL Triangle Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician Blaise Pascal Base on algebra method. only using the coefficients of terms. Power value Coefficient

n = 0 1 n = 1 1 1 n = 2 1 2 1 n = 3 1 3 3 1 n = 4 1 4 6 4 1 n = 5 1 5 10 10 5 1 n = 6 1 ? 1

[email protected]

http://en.wikipedia.org/wiki/Triangular_arrayhttp://en.wikipedia.org/wiki/Binomial_coefficienthttp://en.wikipedia.org/wiki/Trianglehttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://math2ever.blogspot.com


3

Example: (1 + 2x)5

n = 5 1 5 10 10 5 1

(1 + 2x)5 = 5043223405 )2()1(1)2)(1(5)2()1(10)2()1(10)2()1(5)2()1(1 xxxxxx

= xxxxx 32808040101 Method 3 - Binomial theorem Sum of terms (Hasil tambah sebutan)

The general terms = nCr an - rxr

With r = 0,1,2,3,4

How to use calculator to calculate nCr nCr

Example:

Find value of 6C3 using calculator

6 C 3

= 20

nCr =

!!

!

rrn

n

n value SHIFT r value

n value SHIFT r value

=

[email protected]

http://math2ever.blogspot.com


4

The expansion of (a + x)n where n is a positive integer.

The expansion of (a + x)n where n is a positive integer, power for first term a is

n and power for increase terms are (n-1)

example :

n = 5

(a + x)n = an + 5a(n-1)x + 10a(n-2)x2 + 10a(n-3)x3 + 5a(n-4) x4 + x5

(a + x)5 = a5 + 5a4x + 10a3x2 + 10a2x3 + 5ax4 + x5

The general terms of Binomial Expansions:

Formula 1 :

(a + x)n = an + nC1an - 1x + nC2a

n - 2x2 + nC3an - 3x3 + ..+ nCrx

n - rar +

... + xn

= an + na

n - 1x + n(n - 1)a

n - 2x

2 + n(n - 1)(n - 2)a

n- 3x

3 + ... +

1x2 1x2x3

n(n - 1)(n - 2) (n - r + 1)(an - r

xr) + ... + x

n

1x2x3x.... r

[email protected]


5

Example 1: Expand the expression using binomial theorem

a) (1 + 3x)4

b) ( 3 2x )3

download@



6

c) (2 +4

x )5

d) ( x 2y)6

download@



7

e) ( 2a b)4

download@



8

Coefficients value and terms value for positive integer n

Example 2 :

1. Find the coefficient of x3 and 5th term of the binomial expansion of 653 x . a = , x = , n =

General term = nCr an - rxr where r = 0,1,2,3,4,. r = term -1

[email protected]


9

2. Find the term include x3 and the term independent of x in the expansion

12

2

1

xx . a = , x = , n =

Exercise : 1. Expand the expression:

a) 4

21

x

2. Find the 3rd term in the expansion below:

a) 9

2 1

xx b)

62

35

x

3. Find the coefficient value for x18 of expansion 12

33 xx

[email protected]


10

Binomial expansion of (1+x)n where n (power) is not a positive integer

( not positive integer means fraction or negative value )

In this case the Binomial expansions an infinite series.

Condition:

Expansion in form ( 1+x )n

This series is infinite

If expression in form (a + x)n the value of a 1, hence expansion

must use formula 2

General term

If value of a in binomial expression not 1 (a 1) Use formula 2

n

x )1( =

......4321

)3)(2)(1(

321

)2)(1(

21

)1(1

432x

xxx

nnnnx

xx

nnnx

x

nnnx

Expansion a 1

(a + x)n =

n

n

n

a

xa

a

xa

11

n

n

a

xa

1 =

...

4321

)3)(2)(1(

321

)2)(1(

21

)1(1

432

a

x

xxx

nnnn

a

x

xx

nnn

a

x

x

nn

a

xn

[email protected]


11

Example : In ascending power of x, obtain the first 4 terms of :

(a) 1

)2(

x

(b) x4

1

[email protected]


12

(c) 328

2

x

[email protected]


13

d) 22

23

1

x

Exercise : Find the expansion of expression below up to the term in x3 :

a) 31

1 x

b) 1

224

x

c) 21

279 x

[email protected]


14

Approximations

Use Binomial Theorem

If 3)05.2( = + = (2 + 0.05)3

Example :

1. Obtain the approximation value to 4 decimal places

a) (1.08)5 = (1 + 0.08)5 , ( a = x = n = )

use

(a + x)n = an + nC1an - 1x + nC2an - 2x2 + nC3an - 3x3 + ... + nCrxn - rar + ... + xn

(1 + 0.08)5 =

Whole

number x = 2.05 whole number

[email protected]


15

b)

2

997.0

1

[email protected]


16

2. Find the binomial expansion of 521 x . Hence find the value of 50024.1 to

5 places of decimals

[email protected]


17

3. Using Binomial Expansion, find first 4 terms of 102 x . Hence find the

approximation value of 10

98.1 until 4 decimal places.

Expansion Method for approximation

Note :

If value of x is so small or mark slightly

we can use formula

( 1+x )n 1 + nx

exp :

(1+ 0.00002)6 = 1 + 6(0.00002)

= 1 + 0.00012

= 1.00012

(2+ 0.00006)3 = (2 + 0.00006)3

= 23(1 + 3(0.00003))

= 8.00072

[email protected]


18

Approximation for multiplication and division

example :

1. Find the value of expression below to 5 decimal places using binomial

theorem:

i) (1.01)3( 96.0 )

ii)

2

2

1

02.01

06.1

[email protected]


19

iii)

2

3

14

02.01

99.007.1

iv) Find the value of (5.04)4 to four places of decimals using binomial

theorem.

[email protected]

https://math2ever.blogspot.com


20

v) Using binomial theorem, find the value of 7

25

26

to four decimal places.

[email protected]


21

Exercise:

1. Find the binomial expansion of each of the following :

a) 432 yx b) 5

24

x

b) 421 x d)

4

32

x

2. Find the term include x4 and the term independent of x in the expansion 12

3

1

xx .

3. For the Binomial Expansion 16

6

2

2

1

xx

,find coefficient value of x8 .

4. Find the expansion of

3

1

31

12

x

x

5. Expand 31

1 x up to terms x3. Hence using the expansion before, find

the approximation value of 3(1.08) to 5 places of decimals.

6. Find the approximation value for 932.0

1

7. Using binomial theorem, find the value of

3

3

88.0

92.001.1

to four decimal

places.

[email protected]

ba 501 chapter 1 binomial expansions.. sem 5 notes

Documents