Download 2 Binomial Expansions

2 Binomial Expansions
§1. Binomial Expansions for
a Positive Integer
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Binomial Expansion
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(A) By expansion
2
(x+a) =
(x+a)3 =
(x+a)4 =
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In general, if n is a positive integer,
( x  a)
n
 x  C ax
n
n
1
C a x
n
r
r
nr
n
n 1
n 1
C a
n 1
C a x
n
2
2
n2
 ...
 ...
xa
n
• This is called the binomial theorem for a
positive integral index.
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In the expansion of (x+a)n
•
•
•
•
•
•
•
•
(1) there are n +1 terms, beginning with xn
and ending with an,
(2) the sum of the index of a and that of x is
equal to n,
(3)the (r + 1)th term, Cnrarxn-r , is called the
general term of the expansion,
(4) the coefficients Cnr , r = 0,1,2,...., n
are called binomial coefficients. Al
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Example 1
Expand (x+ 2)6 in descending powers of x
Solution
6
6
x +C
5
6
x (2)+C
4
2
6
3
3
=
1
2x (2) +C 3x (2)
+C64 x2(2)4 +C65x(2)5 +(2)6
= x6 +12x5 +60x4 +160x3 +240x2
+192x+64
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Expand (2x-1/4)4 in descending powers of x
•
Solution
1

 2x  
x

4
1
1 3 1 4
4
2 1 2
4
 ( 2 x )  C ( 2 x ) ( )  C 2 ( 2 x ) ( )  C 3 ( 2 x )( )  ( )
x
x
x
x
1 3 1 4
4
3 1
2 1 2
 ( 2 x )  4( 2 x ) ( )  6( 2 x ) ( )  4( 2 x )( )  ( )
x
x
x
x
8 1
4
2
 16 x  20 x  24  2  4
x x
4
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1
3
Binomial Expansion
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• Power
Cofficient
•0
1
•1
1 1
•2
1 2 1
•3
1 3 3 1
•4
1 4 6 4 1
• 5 1 5 10 10 5 1
• This is called Pascal’s triangle.
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(B) The Pascal’s Triangle
• The numerical coefficients in the
binomial expansion :
•
1
•1 1
• 1 C21 1
• 1 C31 C32 1
• 1 C41 C42 C43 1
• 1 C51 C52 C53 C54 1
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•
•
•
•
Cn+1r+1 = Cnr +Cnn+r
C52 = C41+C42
C53= C42+C43 etc.
Thus in the triangle, the end numbers
in each row are each unity
• and the others are the sums of the two
adjacent numbers in the row above.
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Example 3 Expand
(1+x+x2)4 in ascending powers of x.
1  x  x 
2 4
 1  x 1  x 
4
 1  4 x(1  x )  6 x (1  2 x  x ) 
2
2
4 x 3 (1  3 x  3 x 2  x 3 )  x 4 (1  4 x  6 x 2  4 x 3  x 4 )
 1  4 x  4 x 2  6 x 2  12 x 3  6 x 4
 4 x  12 x  12 x  4 x
3
4
5
6
 x  4x  6x  4x  x
4
5
6
7
8
 1  4 x  10 x 2  16 x 3  19 x 4  16 x 5  10 x 6  4 x 7  x 8
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Class Practice 2.2
Expand the following in ascending
powers of x :
•1. (3 + x
3
•2. (3 - 2x )
2
3
•3. (1 + x + x )
4
)
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(D) Find Specific Terms
• Example 4
• (a) Find the coefficient of x3
• (b) the term independent of
x in the expansion of
( x  a )  x  C ax
n
C a x
n
r
r
n
n r
n
1
 ...  a
T ( r  1)  C a x
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r
n r
C a x
n
2
2
n 2
 ...
n
Ans
General .Term
n
r
n 1
3 

 2x  2 
x 

9
:
29×34 , -28×34×7
Binomial Expansion
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(Example 5)
• Expand
2
10
(1 – 2x + 3x ) in
ascending powers of x
3
as far as x .
• (ans) 1 – 20x + 210x2 –1500x3….
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(Example 6)
Find the number of rational
terms in the expansion
( 2  3)
3
100
•(ans) n = 6,12,18,24,….,96
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(Example 7)
Find the coefficient of x3 in the
expanding of
(1+x)3 + (1+x)4 + (1+x)5 +……+
(1+x)10
•(ans) 330
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Class Practice 2.3
• (1) Find
• (a) the coefficient of x2 and
• (b) the term independent of x in
the expansion of
2
(3 x 
3x
)
6
(2) Expand (3-x-x2)3 in the
ascending powers of x as far as x3 .
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(E) Summation Notation
• The sum of terms of a sequence .
n
T
 T1  T2  T3  ....  Tn
r
r 1
Example
100
10
r
2
1


r 5 r  1

r 1
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• The binomial expansion can be
written in summation notation as
n
( x  a)   C a x
n
n
r
r 0
• Formula
n
 (ax
r 1
n
 (x
r 1
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r
nr
n
n
r 1
r 1
 byr )  a  x r  b yr
n
n
n
r 1
r 1
 y r )   x r  2 x r y r   y r
2
r
r
2
r 1
Binomial Expansion
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2
Class Practice 2.4
•
Find the following sum
n
(1) 3(r  3r  2)
2
r 1
100
(2) r (r  1)
r 1
20
100
(3) (2r  1)
r 1
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2
(4) r (r  1)( r  2)
r 1
Binomial Expansion
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• Ex 2a
• Expand the following in ascending
powers of x : (Nos. 1-4)
• (1)
4
• (2) (2x-5y)
• (3) (1-2x)3(1+3x)2
2
4
• (4) (1-2x+x )
6
(1-2x)
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• Expand and simplify : (Nos. 5-6)
8
1  
1 

(5) x     x  
2x  
2x 


 
5
(6) x  1  1 
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
x 1 1
5
21
Simplify : (Nos. 7 – 8)
• (7) (x – y)3 + 3y(x – y)2 + 3y2(x – y) + y3
• (8) (x+1)4 - 4(x+1)3+6(x+1)2 – 4(x+1) +1
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Without using calculators, find the values of : (Nos. 9 – 10)
(9) 1  2  1  2

 


(10)
 

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4
6
5 2  5 2
8 5
Binomial Expansion
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6
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•(11) Find the term independent
of x in the expansion of
 2 1
 3x  
x

6
•(12) Find the ratio of the 6th term to
the 8th term in the expansion of
(2x+3)11 in descending powers of x .
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• (13) Find the coefficient of x2 in the
expansion of (1 - 3x+x2)7.
• (14) Find the coefficient of x3 in the
expansion of (1 - x)(1+2x)4.
• (15) Find the coefficient of x3 in the
expansion of (1+2x)9 - (1 - 2x)11.
•
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• (16) Find the coefficient of x2 in the
expansion of (1 - x+x2)n .
• (17) Find the first three terms in the
expansion of (1 - 2x)(1+2x3)8 in
ascending powers of x. What is the
coefficient of x10 ?
• (18) Find the values of p and n if the
first three terms in the expansion of
(1+px)n are 1 - 24x+252x2.
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• (19) Write down the (r + 1)th term
in the expansion of

3
x
2
x

11
in ascending powers of x.
If the power of x in this term is
an integer, find the possible values
of r.
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• (20) If the coefficient of x2 in the
expansion of (1+x+x2)n is 21 and n is
a positive integer, find the value of n.
• (21) In the expansion of (x2 + 2)n in
descending powers of x, where n is a
positive integer, the coefficient of the
third term is 40. Find the value of n
and the coefficient of x4.
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• (22) Expand (1+ax)4 (1 - 4x)3 in
ascending powers of x up to
and including the term
2
containing x .
Given that the coefficient of x
is zero, evaluate the coefficient
of x2.
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Binomial Expansion
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n
1

2
1  x  x 
2


• in ascending powers of x, up to and
including the term in x3,
simplifying the coefficients.
• Verify that when n = 9 the
coefficient of x2 is zero, and find the
value of the coefficient of x3 in this
case.
• (23) Expand
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• (24) Find a if the coefficient of x
in the expansion of
(l+ax)8(l +3x)4 - (1+x)3(l+2x)4
is zero. What is the coefficient
of x2 ?
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• (25) Give the first 4 terms in the binomial
expansion of (1 + ax)n in ascending
powers of x. Show that the ratio of the
coefficient of xr+1 to that of xr is
a(n  r )
( r  1)
• Given that the ratio of the coefficient of
x6 to that of x5 is 30, and that the ratio of
the coefficient of x9 to that of x8 is 15, find
integral values for a and n.
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