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
o
o
o
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