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ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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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
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http://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Binomialhttp://math2ever.blogspot.com
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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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
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
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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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
=
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ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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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
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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Example 1: Expand the expression using binomial theorem
a) (1 + 3x)4
b) ( 3 2x )3
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ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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c) (2 +4
x )5
d) ( x 2y)6
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ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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e) ( 2a b)4
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ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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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
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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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
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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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
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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Example : In ascending power of x, obtain the first 4 terms of :
(a) 1
)2(
x
(b) x4
1
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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(c) 328
2
x
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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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
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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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
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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b)
2
997.0
1
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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2. Find the binomial expansion of 521 x . Hence find the value of 50024.1 to
5 places of decimals
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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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
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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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
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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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.
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ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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v) Using binomial theorem, find the value of 7
25
26
to four decimal places.
ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501
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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.