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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 overview

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 $ n=2,3$, 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:


[Annotations]

  automatically generated 9/18/2007