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The '''binomial theorem''', also known as the binomial expansion, describes a method for expanding brackets containing two terms raised to a power as a [[polynomial]].<ref name=mathworld>[http://mathworld.wolfram.com/BinomialTheorem.html Binomial theorem] from mathworld.wolfram.com</ref> More specifically, it describes how to express a bracket of the form (x+a)<sup>n</sup> as a polynomial with terms b<sub>k</sub>x<sup>k</sup>a^<sup>n-k</sup>, for various coefficients b<sub>k</sub>. When n is [[natural number]], the binomial theorem produces a finite length [[polynomial]], but can also be extended to non-integer n. For n not being a natural number, the binomial expansion of (x+a)<sup>n</sup> is:<ref name=mathworld/>
:<math>
(x+a)^n = \sum^n_{k=0} {n \choose k} x^k a^{n-k}
</math>
Here, <math>n \choose k</math> is the binomial coefficient and pronounced "n choose k". It is derived from [[combinatorics]] and equal to:
:<math>
{n \choose k} = \frac{n!}{(n-k)!k!}
</math>
These coefficients for constant n form the rows of [[Pascal's triangle]]. The name comes from the [[Latin]], bi-nomin, meaning two names (terms).<ref>[https://www.merriam-webster.com/dictionary/binomial Etymology] from Merriam-Webster.com</ref>
==Proof==
The binomial theorem for integer n can easily be proved using [[proof by induction]].<ref name=MathematicalMethods>K.F. Riley, M.P. Hobson, S.J. Bence, ''Mathematical Methods for Physics and Engineering'', Cambridge University Press, 3<sup>rd</sup> ed., 2006</ref> Supposing the theorem is true for n=N. Then:
:<math>
(x+a)^{N+1} = (x+a) \sum^N_{k=0} {N \choose k} x^k a^{N-k}
</math>
Multiplying out the bracket, and substituting j=k+1 gives:
:<math>
(x+a)^{N+1} = \sum^N_{k=0} {N \choose k} x^k a^{N+1-k} + \sum^{N+1}_{j=1} {N \choose j-1} x^j a^{N+1-k}
</math>
Separating the k=0 and j=N+1 terms, the sums can be combined as:
:<math>
(x+a)^{N+1} = a^{(N+1)} + \sum^N_{k=1} \left( {N \choose k}+{N \choose k-1} \right) x^k y^{(N+1)-k} + {N+1 \choose N+1}x^{N+1}
</math>
Using a combinations identity, that <math>{N \choose k}+{N \choose k-1}={N+1 \choose k}</math>,<ref name=MathematicalMethods/> this can be rewritten as:
:<math>
(x+a)^{N+1} = \sum^{N+1}_{k=0} {N+1 \choose k} x^k a^{N-k}
</math>
Hence if the theorem holds for n=N, it must hold for n=N+1. As it is true for n=0, it is true for all [[natural numbers]].
==Generalisation to Non-Integers==
When the binomial theorem is extended to non-integer n or n<0, the series it creates is [[infinite]]:<ref name=mathworld/>
:<math>
(x+a)^n = \sum^{\infty}_{k=0} {n \choose k} x^k a^{n-k} = a^n + nxa^{n-1} + \frac{n(n-1)}{2!} x^2 a^{n-2} + \frac{n(n-1)(n-2)}{3!} x^3 a^{n-3}...
</math>
As it is an infinite series, it is often referred to as the "binomial series". The binomial coefficients are generalised so that n need not be a [[natural number]] as:<ref>[http://tutorial.math.lamar.edu/Classes/CalcII/BinomialSeries.aspx Binomial series] from tutorial.math.lamar.edu</ref>
:<math>
{n \choose k} = \frac{1}{k!} \prod^{k}_{i=1}(n-k+1) = \frac{n(n-1)(n-2)...(n-k+1)}{k!}
</math>
for k>0 and 1 for k=0. As this series contains an infinite number of terms, one must worry whether it [[Convergence|converges]] or not. In the case of the binomial expansion, it always converges if |x/a|<1 or if n a [[natural number]].<ref name=mathworld/>
Other generalisations exist such as the multinomial expansion for brackets with more than two terms and the multi-binomial expansion where several binomial brackets are multiplied together.
==Example==
As an example, consider expanding (x+y)<sup>4</sup>. It would take a while doing it by hand. Using the binomial theorem, we quickly derive:
:<math>
(x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 +y^4
</math>
These coefficients form the 5th row of Pascal's triangle. As an example with negative n, consider (x+1)<sup>-1</sup>:
:<math>
(x+1)^{-1} = 1 - x + \frac{1}{2} x^2 - \frac{1}{6} x^3 + \frac{1}{24} x^4 -...
</math>
which will converge for |x|<1.
==References==
{{reflist}}
==See also==
*[[Polynomial]]
*[[Combinatorics]]
*[[Pascal's triangle]]
[[Category:Mathemtatics]]
[[Category:Algebra]]
:<math>
(x+a)^n = \sum^n_{k=0} {n \choose k} x^k a^{n-k}
</math>
Here, <math>n \choose k</math> is the binomial coefficient and pronounced "n choose k". It is derived from [[combinatorics]] and equal to:
:<math>
{n \choose k} = \frac{n!}{(n-k)!k!}
</math>
These coefficients for constant n form the rows of [[Pascal's triangle]]. The name comes from the [[Latin]], bi-nomin, meaning two names (terms).<ref>[https://www.merriam-webster.com/dictionary/binomial Etymology] from Merriam-Webster.com</ref>
==Proof==
The binomial theorem for integer n can easily be proved using [[proof by induction]].<ref name=MathematicalMethods>K.F. Riley, M.P. Hobson, S.J. Bence, ''Mathematical Methods for Physics and Engineering'', Cambridge University Press, 3<sup>rd</sup> ed., 2006</ref> Supposing the theorem is true for n=N. Then:
:<math>
(x+a)^{N+1} = (x+a) \sum^N_{k=0} {N \choose k} x^k a^{N-k}
</math>
Multiplying out the bracket, and substituting j=k+1 gives:
:<math>
(x+a)^{N+1} = \sum^N_{k=0} {N \choose k} x^k a^{N+1-k} + \sum^{N+1}_{j=1} {N \choose j-1} x^j a^{N+1-k}
</math>
Separating the k=0 and j=N+1 terms, the sums can be combined as:
:<math>
(x+a)^{N+1} = a^{(N+1)} + \sum^N_{k=1} \left( {N \choose k}+{N \choose k-1} \right) x^k y^{(N+1)-k} + {N+1 \choose N+1}x^{N+1}
</math>
Using a combinations identity, that <math>{N \choose k}+{N \choose k-1}={N+1 \choose k}</math>,<ref name=MathematicalMethods/> this can be rewritten as:
:<math>
(x+a)^{N+1} = \sum^{N+1}_{k=0} {N+1 \choose k} x^k a^{N-k}
</math>
Hence if the theorem holds for n=N, it must hold for n=N+1. As it is true for n=0, it is true for all [[natural numbers]].
==Generalisation to Non-Integers==
When the binomial theorem is extended to non-integer n or n<0, the series it creates is [[infinite]]:<ref name=mathworld/>
:<math>
(x+a)^n = \sum^{\infty}_{k=0} {n \choose k} x^k a^{n-k} = a^n + nxa^{n-1} + \frac{n(n-1)}{2!} x^2 a^{n-2} + \frac{n(n-1)(n-2)}{3!} x^3 a^{n-3}...
</math>
As it is an infinite series, it is often referred to as the "binomial series". The binomial coefficients are generalised so that n need not be a [[natural number]] as:<ref>[http://tutorial.math.lamar.edu/Classes/CalcII/BinomialSeries.aspx Binomial series] from tutorial.math.lamar.edu</ref>
:<math>
{n \choose k} = \frac{1}{k!} \prod^{k}_{i=1}(n-k+1) = \frac{n(n-1)(n-2)...(n-k+1)}{k!}
</math>
for k>0 and 1 for k=0. As this series contains an infinite number of terms, one must worry whether it [[Convergence|converges]] or not. In the case of the binomial expansion, it always converges if |x/a|<1 or if n a [[natural number]].<ref name=mathworld/>
Other generalisations exist such as the multinomial expansion for brackets with more than two terms and the multi-binomial expansion where several binomial brackets are multiplied together.
==Example==
As an example, consider expanding (x+y)<sup>4</sup>. It would take a while doing it by hand. Using the binomial theorem, we quickly derive:
:<math>
(x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 +y^4
</math>
These coefficients form the 5th row of Pascal's triangle. As an example with negative n, consider (x+1)<sup>-1</sup>:
:<math>
(x+1)^{-1} = 1 - x + \frac{1}{2} x^2 - \frac{1}{6} x^3 + \frac{1}{24} x^4 -...
</math>
which will converge for |x|<1.
==References==
{{reflist}}
==See also==
*[[Polynomial]]
*[[Combinatorics]]
*[[Pascal's triangle]]
[[Category:Mathemtatics]]
[[Category:Algebra]]