BINOMIAL THEOREM
Frequently Asked Questions
The Binomial Theorem gives the expansion of \((a+b)^n\), where \(n\) is a non-negative integer, in the form \(\sum_{r=0}^{n} {n \choose r} a^{n-r} b^r\).
The general term (r+1)th term is \(T_{r+1} = {n \choose r} a^{n-r} b^r\).
A binomial expression is an algebraic expression consisting of exactly two unlike terms, such as \(a+b\) or \(x-2y\).
The binomial coefficient \({n \choose r}\) represents the number of ways of choosing \(r\) objects from \(n\) objects and equals \(\dfrac{n!}{r!(n-r)!}\).
The theorem applies when the exponent is a non-negative integer.
\((a+b)^2 = a^2 + 2ab + b^2\).
\((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\).
There are \(n+1\) terms in the expansion.
If \(n\) is even, the middle term is the \(\left(\dfrac{n}{2}+1\right)\)th term; if \(n\) is odd, there are two middle terms.
The middle term is the 4th term: \(T_4 = {6 \choose 3}x^3y^3\).
Pascal’s Triangle is a triangular arrangement of binomial coefficients.
\({n \choose r} = {n-1 \choose r} + {n-1 \choose r-1}\).
The first term is \(a^n\).
The last term is \(b^n\).
The coefficient of the general term is \({n \choose r}\).