The Binomial Theorem. Expansion of Binomials Binomial Expansion Coefficients Pascal’s Triangle...
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The Binomial Theorem
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Transcript of The Binomial Theorem. Expansion of Binomials Binomial Expansion Coefficients Pascal’s Triangle...
The Binomial Theorem
Expansion of Binomials
0
1
2 2 2
3 3 2 2 3
4 4 3 2 2 3 4
5 5 4 3 2 2 3 4 5
( ) 1
( )
( ) 2
( ) 3 3
( ) 4 6 4
( ) 5 10 10 5
x y
x y x y
x y x xy y
x y x x y xy y
x y x x y x y xy y
x y x x y x y x y xy y
Binomial Expansion Coefficients
Pascal’s Triangle
Row
1 0
1 1 1
1 2 1 2
1 3 3 1 3
1 4 6 4 1 4
1 5 10 10 5 1 5
Binomial Coefficient
nr
nC
r
n-Factorial or n!
n-Factorial
For any positive integer n,
and
! ( 1)( 2) (3)(2)(1),
0! 1 .
n n n n
Example Evaluate (a) 5! (b) 7!
Solution (a)
(b)
5! 5 4 3 2 1 120
7! 7 6 5 4 3 2 1 5040
Simplifying r!
2!
3!
4!
3!
! ( 1)( 2)...3 2 1r r r r
7! 7 6 5 4 3 2 1
!
!
n
n
4
11
3
Simplifying
( 2)!
( 3)!
n
n
( 5)!
( 4)!
n
n
( 2)n
1
( 4)n
Simplifying
3
4
2
2
n
n
3
2
2
2 2
2
Binomial Coefficient
• The symbols and for the binomial
coefficients are read “n choose r”
• The values of are the values in the nth
row of Pascal’s triangle.
So is the first number in the third row
and is the third.
n rCn
r
n
r
3
0
3
2
Binomial Coefficient
Binomial Coefficient
For non-negative integers n and r, with r < n,
!
!( )!n
r
n nC
r r n r
Formulae
!
! !
n n
r r n r
( 1)( 2)...( 1)
!
n n n n n r
r r
Factorial Formula
Multiplicative Formula
Evaluating Binomial Coefficients
Example Evaluate (a) (b)
Solution
(a)
(b)
6
2
8
0
6 6! 6! 6 5 4 3 2 115
2 2!(6 2)! 2!4! 2 1 4 3 2 1
8 8! 8! 8!1
0 0!(8 0)! 0!8! 1 8!
The Binomial Theorem
Binomial Theorem
For any positive integers n,
1 2 2 3 3
1
( )1 2 3
... ...1
n n n n n
n r r n n
n n nx y x x y x y x y
n nx y xy y
r n
Applying the Binomial Theorem
Example Write the binomial expansion of .
Solution Use the binomial theorem
9( )x y
9 9 8 7 2 6 3
5 4 4 5 3 6 2 7
8 9
9 9 9( )
1 2 3
9 9 9 9
4 5 6 7
9
8
x y x x y x y x y
x y x y x y x y
xy y
Applying the Binomial Theorem
9 9 8 7 2 6 3
5 4 4 5 3 6 2 7
8 9
9 8 7 2 6 3 5 4 4 5
3 6 2 7 8 9
9! 9! 9!( )
1!8! 2!7! 3!6!9! 9! 9! 9!
4!5! 5!4! 6!3! 7!2!9!
8!1!
9 36 84 126 126
84 36 9
x y x x y x y x y
x y x y x y x y
xy y
x x y x y x y x y x y
x y x y xy y
Applying the Binomial Theorem
Example Expand .
Solution Use the binomial theorem with
and n = 5,
5
2
ba
2 35 5 4 3 2
4 5
5 5 5( )
1 2 32 2 2 2
5
4 2 2
b b b ba a a a a
b ba
,2
bx a y
Applying the Binomial Theorem
Solution
2 35 5 4 3 2
4 5
5 4 3 2 2 3 4 5
( ) 5 10 102 2 2 2
52 2
5 5 5 5 1
2 2 4 16 32
b b b ba a a a a
b ba
a a b a b a b ab b
r th Term of a Binomial Expansion
rth Term of the Binomial Expansion
The rth term of the binomial expansion of (x + y)n,
where n > r – 1, is
( 1) 1
1n r rnx y
r
Finding a Specific Term of a Binomial Expansion.
Example Find the fourth term of .
Solution Using n = 10, r = 4, x = a, y = 2b in the
formula, we find the fourth term is
10( 2 )a b
7 3 7 3 7 310(2 ) 120 8 960 .
3a b a b a b
Pg 130 E.g 11(a)Find, in ascending powers of x, the 1st 4 terms of
the expansion (1+2x)n, where n > 2. Given that the coefficients of x3 and x2 are in the ratio 14 : 3, find n.
( 1)( 2)...( 1)
!
n n n n n r
r r
Using Multiplicative Formula:
2 3(1 2 ) 1 (2 ) (2 ) (2 ) ...1 2 3
n n n nx x x x
3
2
(2)3 14
3(2)
2
n
n
2 ( 1)( 2) ( 1) 14
3! 2! 3
n n n n n
2 ( 1)( 2) 2! 14
3! ( 1) 3
n n n
n n
2( 2) 14
3 32 7
9
n
n
n
Using Factorial Formula:
!
! !
n n
r r n r
! ! 14
( 3)!3! ( 2)!2! 3
n n
n n
! ( 2)!2! 14
( 3)!3! ! 3
n n
n n
2( 2) 14
3 32 7
9
n
n
n
2
2
Pg 132 Q9(a)In the expansion of (2+3x)n, the coefficients of x3
and x4 are in the ratio 8 : 15. Find n.
1 2 2 3 3 4 4(2 3 ) 2 2 (3 ) 2 (3 ) 2 (3 ) 2 (3 ) ...1 2 3 4
n n n n n nn n n nx x x x x
3 3
4 4
2 (3)3 8
152 (3)
4
n
n
n
n
Using Factorial Formula: !
! !
n n
r r n r
3 3 4 4!2 (3) !2 (3) 8
( 3)!3! ( 4)!4! 15
n nn n
n n
8 8
3( 3) 15
3 5
8
n
n
n
3 3
4 4
!2 (3) ( 4)!4! 8
( 3)!3! !2 (3) 15
n
n
n n
n n
