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BodhiAI
Nov. 12, 2019

Binomial Theorem & Types:

Binomial Expression

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 

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 

Introduction

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