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# Binomial Theorem by 01ySz72V

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```									                                                                                                     FM 30(IB)
The Binomial Theorem (supplementary)
Pascal’s Triangle
1
1 1               What is the pattern?
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1

    What will be the elements of the next row?

The Theorem

( a  b) 0                                                    1
(a  b)1                                                     ab
( a  b) 2                                               a 2  2ab  b2
( a  b)3                                            a3  3a 2 b  3ab2  b3
( a  b) 4                                       a 4  4a3b  6a 2 b2  4ab3  b4
( a  b)5                                    a5  5a 4 b  10a3b2  10a 2 b3  5ab4  b5

**Note:         If you consider (a  b)n ,
       the 1st term is a n
       the last term is b n
       the exponents of ‘a’ decrease from left to right, while the exponents of ‘b’ increase
       the sum of the exponents in each term = n

 How is Pascal’s Triangle related to the Binomial Theorem?

 In what row of Pascal’s Triangle would you find the coefficients of (a  b)10 ?

The theorem is…
(a  b)n n C0 an n C1 an1 b n C2 an2b2 n C3 an3b3  ...n Cnbn

Isn’t it beautiful????? Don’t memorize it – look at the pattern…
 We use combinations to find the coefficients!

Examples: Develop. Feel the movement in your heart.
1. ( x  y )6
2. (2 x3  3 y 2 ) 4

To Find the ‘r-th’ term…
If ‘r ‘ is the position of the term, le r-th term of the development of (a  b)n is

n Cr 1     an(r 1) br 1

Example
11
      1 
1. Find the 6th term of ( x  4 y )12 .                                 2. Determine the 7th term of  x5  2  .
     x 

r  6 
a  x 
          
 b  4 y 
          
 n  12 

Homework:
1. Expand.
a)  x  2 y                       2 x  1                  x        1                2x        3 y3 
6                              6                           5                                    4
b)                            c)        3
d)          2

2. Find the r-th term.
a) 7th term of  a  b                 b) 4th term of  x  3 y               c) 15th term of  x  y 
10                                               8                                            20

d) 5th term of  2 x  3 y             e) 4th term of (2a 2  3 xy 3 ) 6
7

2

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