Binomial theorem: Difference between revisions
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In [[elementary algebra]], the '''binomial theorem''' or the binomial expansion is the identity that states that for any non-negative [[integer]] ''n'', | In [[elementary algebra]], the '''binomial theorem''' or the binomial expansion is a mechanism by which expressions of the form <math>(x + y)^n</math> can be expanded. It is the identity that states that for any non-negative [[integer]] ''n'', | ||
: <math> (x + y)^n = \sum_{k=0}^n {n \choose k} x^k y^{n-k}, </math> | : <math> (x + y)^n = \sum_{k=0}^n {n \choose k} x^k y^{n-k}, </math> |
Revision as of 07:15, 9 August 2010
In elementary algebra, the binomial theorem or the binomial expansion is a mechanism by which expressions of the form can be expanded. It is the identity that states that for any non-negative integer n,
or, equivalently,
where
is a binomial coefficient.
One way to prove this identity is by mathematical induction.
Proof:
Base case: n = 0
Induction case: Now suppose that it is true for n : and prove it for n + 1.
and the proof is complete.
The first several cases
Newton's binomial theorem
There is also Newton's binomial theorem, proved by Isaac Newton, that goes beyond elementary algebra into mathematical analysis, which expands the same sum (x + y)n as an infinite series when n is not an integer or is not positive.