Difference between revisions of "Binomial Theorem"
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Revision as of 12:09, 12 July 2006
The Theorem
First discovered by Newton, the Binomial Theorem states that for real or complex a,b,
.
This may be shown for the integers easily:
.
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 .
Usage
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, and knowing the first few coefficients would also be beneficial.