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,
x) If and
denote respectively the sum of odd terms and even terms in the expansion of
, then
(a) and
(b)
(c)
(d)