# 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,

 $(a+b)^{n}=a^{n}+\binom{n}{1}a^{n-1}b+\binom{n}{2}a^{n-2}b^{2}+\cdots+b^{n}.$

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

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

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