Mathematics-Online lexicon: Binomial Theorem

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	Mathematics-Online lexicon:
	Binomial Theorem

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

The binomial identity provides explicit expressions for integral powers of a sum of two variables:

$\displaystyle (a+b)^n$	$\displaystyle =$	$\displaystyle a^n + \left( \begin{array}{c} n \\ 1 \end{array}\right) a^{n-1}b ... ...^2 + \cdots + \left( \begin{array}{c} n \\ n-1 \end{array}\right)ab^{n-1} + b^n$
	$\displaystyle =$	$\displaystyle \sum_{k=0}^n \left( \begin{array}{c} n \\ k \end{array}\right)a^{n-k}b^k \,,$

for all $n\in\mathbb{N}_0$ .

In particular, for , the formula yields

$\displaystyle (a+b)^2$	$\displaystyle =$	$\displaystyle a^2+2ab+b^2\,,$
$\displaystyle (a+b)^3$	$\displaystyle =$	$\displaystyle a^3+3a^2 b+3ab^2 + b^3\,.$

(Authors: Kimmerle/Abele)

see also:

automatically generated 9/18/2007