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binomial theorem (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,

$\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 $ n$ 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 $ a$ and $ b$ belong to a commutative rig.



"binomial theorem" is owned by KimJ.
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See Also: binomial formula, binomial coefficient, binomial distribution, using the primitive element of biquadratic field

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:
AMS MSC11B65 (Number theory :: Sequences and sets :: Binomial coefficients; factorials; $q$-identities)

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generality by rspuzio on 2005-02-22 04:18:26
It might be worth pointing out that this theorem also holds if a and b belong to an commutative rig (the spelling is right; I really mean "rig", not "ring" here) since we only use some basic algebraic properties of real or complex numbers in the proof.
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