Binomial Theorem
We know how to find the squares and cubes of binomials like a + b and a – b . E . g . ( a + b ) 2 , ( a-b ) 3 etc . However , for higher powers calculation becomes difficult . This difficulty was overcome by a theorem known as binomial theorem . It gives an easier way to expand ( a + b ) n , where n is an integer or a rational number .
Points to note in Binomial Theorem
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Total number of terms in expansion = index count + 1 . g . expansion of ( a + b ) 2 , has 3 terms .
Powers of the first quantity ‘ a ’ go on decreasing by 1 whereas the powers of the second quantity ‘ b ’ increase by 1 , in the successive terms .
In each term of the expansion , the sum of the indices of a and b is the same and is equal to the index of a + b .
Numerical : Compute ( 98 ) 5 Solution : ( 98 ) 5 = ( 100-2 ) 5 =
= 5 C0 ( 100 ) 5 – 5 C1 ( 100 ) 4 . 2 + 5 C2 ( 100 ) 3 2 2 – 5 C3 ( 100 ) 2 ( 2 ) 3 + 5 C4 ( 100 ) ( 2 ) 4 – 5 C5 ( 2 ) 5 = 10000000000 – 5 × 100000000 × 2 + 10 × 1000000 × 4 – 10 × 10000 × 8 + 5 × 100 × 16 – 32
= 10040008000 – 1000800032 = 9039207968 General term in the expansion of ( a + b ) n is Tr + 1 = n Cr a n – r b r Numerical : Find the 4 th term in the expansion of ( x – 2y ) 12
Solution : Putting r = 3 , n = 12 , a = x & b = -2y in this formula the formula , Tr + 1 = n Cr a n – r b r