Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag
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 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)

Annotation:

[Links]
  automatically generated 9/18/2007