Binomial Theorem
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 , I would 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.