What is binomial expansion theorem?

Answer 1

We often come across the algebraic identity (a + b)2 = a2 + 2ab + b2. In expansions of smaller powers of a binomial expressions, it may be easy to actually calculate by working out the actual product. But with higher powers the work becomes very cumbersome.

The binomial expansion theorem is a ready made formula to find the expansion of higher powers of a binomial expression.

Let ( a + b) be a general binomial expression. The binomial expansion theorem states that if the expression is raised to the power of a positive integer n, then,

(a + b)n = nC0an + nC1an-1 b+ nC2an-2 b2+ + nC3an-3 b3+ ………+ nCn-1abn-1+ + nCnbn

The coefficients in each term are called as binomial coefficients and are represented in combination formula. In general the value of the coefficient

nCr = n!r!(n-r)!

It may be interesting to note that there is a pattern in the binomial expansion, related to the binomial coefficients. The binomial coefficients at the same position from either end are equal. That is,

nC0 = nCn nC1 = nCn-1 nC2 = nCn-2 and so on.

The advantage of the binomial expansion theorem is any term in between can be figured out without even actually expanding.

Since in the binomial expansion the exponent of b is 0 in the first term, the general term, term is defined as the (r+1)th b term and is given by Tr+1 = nCran-rbr

The middle term of a binomial expansion is [(n/2) + 1]th term if n is even. If n is odd, then terewill be two middle terms which are [(n+1)/2]th and [(n+3)/2]th terms.