Binomial Expression: An expression consisting of two terms, connected by or sign is called binomial expression.

Binomial Theorem: If and are real numbers then for all , we have

This expression has the following properties:

i) It has terms

ii) The sum of the indices of and in each terms is n

iii) The coefficients of terms equidistant from the beginning and the end are equal.

iv) The general term

v) can also be expressed as

vi) Replacing by in the expression , we get

The general term in the expansion of is

vii) Putting and replacing by , in the expression we get

viii) Putting in the expression , we get

This is the expansion of in descending powers of . In this case,

ix) Addition and Subtraction

Similarly,

If is odd, then and both have terms.

If is even, then has terms whereas both have terms.

x) If and denote respectively the sum of odd terms and even terms in the expansion of , then

(a) and

(b)

(c)

(d)

xi) If is even, then term is the middle term.

If is off, then and are middle terms

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