Difference between revisions of "Binomial Theorem"
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Revision as of 11:51, 30 April 2008
The Binomial Theorem states that for real or complex , , and non-negative integer ,
This may be easily shown for the integers: . Repeatedly using the distributive property, we see that for a term , we must choose of the terms to contribute an to the term, and then each of the other terms of the product must contribute a . Thus, the coefficient of is . Extending this to all possible values of from to , we see that .
The Binomial Theorem was generalized by Isaac Newton, who used an infinite series to allow for complex exponents: For any real or complex , , and ,
Many factorizations involve complicated polynomials with binomial coefficients. For example, if a contest problem involved the polynomial , one could factor it as such: . It is a good idea to be familiar with binomial expansions, including knowing the first few binomial coefficients.