Download Binomial Theorem (Pascal`s Triangle)

MA: Binomial Expansion (1st technique: Pascal’s Triangle)
1
1
1
1
1
__
2
3
4
__
1
1
3
6
__
1
4
__
1
__
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These numbers are the coefficients for each term in a binomial expansion.
(a  b)0  1a0b0
(a  b)1  1a1b0  1a0b1
(a  b)2  1a 2b0  2a1b1  1a 0b2
(a  b)3  1a3b0  3a 2b1  3a1b2  1a0b3
(a  b)4  1a 4b0  4a3b1  6a 2b2  4a1b3  1a 0b4
What will the next expansion be?
(a  b)5 
Use the pattern in Pascal’s triangle to write ( x  y)7 in expanded form.
Binomial Expansion (2nd technique: Binomial Theorem)
Before beginning we need to know the notation C (known as a Combination)
n r
n
Cr 
n!
 n  r  !r !
Example: Find the value of C :
11
5 3
C5 :
The Binomial Theorem:
n
In the expansion of (x + y)
n
n
n–1
(x + y) = x +nx
n–r r
n–1
y + … + C x y + … + nxy
n r
n
+y
n–r r
The coefficient of x y is
n
Cr 
n!
 n  r  !r !
n
The symbol   is often used in place of C to denote
n r
r 
binomial coefficients.
Ex 2: Expand ( x  y )3 using the Binomial Theorem:
( x  y )3 = 3 C0 x3 
Ex 3: Expand (2 x  y)5 using the Binomial Theorem:
Finding a certain term of a binomial expansion:
Example: Find the 4th term of ( x  2 y)6 :
Here n = 6. To find r, use: (term number desired) – 1. Therefore r = 4-1 = 3.
6
C3 x 63 (2 y )3 
1) Find the 3rd term of (2 x  y)6 :
2) Find the 5th term of ( x  1)9 :