binomial theorem

The binomial theorem is a formula for the expansion of ${(a+b)}^n$, for $n$ a positive integer and $a$ and $b$ any two real (or complex) numbers, into a sum of powers of $a$ and $b$. More precisely,

$${(a+b)}^n=a^n+\left(\genfrac{}{}{0pt}{}{n}{1}\right)a^{n-1}b+\left(\genfrac{}{}{0pt}{}{n}{2}\right)a^{n-2}{b}^2+\mathrm{\cdots }+b^n.$$

For example, if $n$ is 3 or 4, we have:

${(a+b)}^3$ $=a^3+3a^{2}b+3a{b}^2+b^3$

${(a+b)}^4$ $=a^4+4a^{3}b+6a^2{b}^2+4a{b}^3+b^4.$

This result actually holds more generally if $a$ and $b$ belong to a commutative rig (http://planetmath.org/Rig).

Title	binomial theorem
Canonical name	BinomialTheorem
Date of creation	2013-03-22 11:46:47
Last modified on	2013-03-22 11:46:47
Owner	KimJ (5)
Last modified by	KimJ (5)
Numerical id	17
Author	KimJ (5)
Entry type	Theorem
Classification	msc 11B65
Classification	msc 06F25
Classification	msc 03E20
Related topic	BinomialFormula
Related topic	BinomialCoefficient
Related topic	BernoulliDistribution2
Related topic	UsingThePrimitiveElementOfBiquadraticField