Binomial Theorem
The Binomial Theorem states that for real or complex ,
, and non-negative integer
,
![$(a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$](http://latex.artofproblemsolving.com/4/9/4/494769588adb2ac3700e25b086fe2b7d41bba70a.png)
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
.
Generalization
The Binomial Theorem was generalized by Isaac Newton, who used an infinite series to allow for complex exponents: For any real or complex ,
, and
,
![$(a+b)^r = \sum_{k=0}^{\infty}\binom{r}{k}a^{r-k}b^k$](http://latex.artofproblemsolving.com/0/8/9/0896503fb81e2e64fc5d22e03210b9fc46b7ce32.png)
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, including knowing the first few binomial coefficients.