An algebraic expression consisting of only two terms is called a Binomial Expression.
Example: (i) a + b (ii) 3a + 4b (iii) a2-b2 (iv)√x +k/x2
Binomial theorem is the formula obtained by which the binomial expression is raised to the power of n, where n can be positive, negative or a fraction.
Types of Binomial Theorem
There are two types of Binomial Theorem
a) Binomial Theorem for Positive Integral Index
b) Binomial Theorem for Negative/Rational Index.
Binomial Theorem for Positive Integral Index
It states that, if n is any positive integer, then the expansion of (x+a)n can be written as
Reasoning: As we can see, in the expression (x+a)n, we have actually n factors of (x+a), such that
It can be easily sai that the above expression on expansion will be an n degree equation in x and will contain all the coefficients of (constant).
General Term The number of ways in which (n-r) x's and r a's can be taken out from n factors
(r+1)th term are will occur times in the product. Hence, (r+1)th term of the series will be
Last Term The number of ways in which ZERO x's and n a's can be taken out from n factors
(n+1)th term are will occur times in the product. Hence, (n+1)th term of the series will be
Combining all the terms together, the binomial expansion can be written as