PlanetMath: binomial theorem

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binomial theorem

(Theorem)

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

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

For example, if

is 3 or 4, we have:

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

This result actually holds more generally if and belong to a commutative rig.

"binomial theorem" is owned by KimJ.

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Keywords:

number theory combinatorics

Attachments:: inductive proof of binomial theorem (Proof) by Mathprof
proof of binomial theorem (Proof) by mps

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Cross-references: commutative, sum of powers, numbers, complex, real, integer, positive
There are 26 references to this entry.

This is version 12 of binomial theorem, born on 2001-10-16, modified 2005-02-22.
Object id is 247, canonical name is BinomialTheorem.
Accessed 20065 times total.

Classification:

11B65 (Number theory :: Sequences and sets :: Binomial coefficients; factorials; $q$-identities)

Pending Errata and Addenda

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