Binomial Theorem
Revision as of 12:52, 22 June 2006 by MCrawford (talk | contribs) (Binomial theorem moved to Binomial Theorem: proper noun capitalization)
First invented by Newton, the Binomial Theorem states that for real or complex a,b,
This may be shown for the integers easily:
![$\displaystyle (a+b)^n=\underbrace{ (a+b)\cdot(a+b)\cdot(a+b)\cdot\cdots\cdot(a+b) }_{n}$](http://latex.artofproblemsolving.com/a/9/d/a9de53f8bebe423d8614c4e58a9f1a0fb5244b30.png)
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
.