Is there a polynomial equation for $f(n) = n!$ and if so what is it? And I am not necessarily talking about $f(n) = n(n-1)(n-2)...(3)(2)(1)$ in its factored form; Well it could be that but then I would like a general way of expansion.
Thanks in advance!
 A: You can always write a polynomial $P(x)$ in the following form : 
$$
P(x) = \sum_{i=0}^m a_i x^i = x^m \left( \sum_{i=0}^m a_i x^{i-m} \right).
$$
When you plug in an integer $n$, you get
$$
\frac{P(n)}{n^m} = \sum_{i=0}^m a_i n^{i-m} \to a_m \neq 0.
$$
Therefore, for any polynomial $P$ of degree $m$, we have
$$
\frac{P(n)}{n!} = \frac{P(n)}{n^m} \frac{n^m}{n!} \to a_m \lim_{n \to \infty} \frac{n^m}{n!}.
$$
You can easily check that the last limit is zero for all positive integers $m$. If $n!$ would be a polynomial, $P(n)/n!$ would be identically $1$ for some choice of polynomial $P$, a contradiction.
Hope that helps,
A: There is no polynomial $P(n)$ such that $n!=P(n)$ for all $n$: the factorial function grows faster than any polynomial.
One way of showing this is to note that $n!\gt 2^n$ if $n\ge 4$. Then we can use L'Hospital's Rule repeatedly to show that if $P(x)$ is any polynomial, then $\lim_{x\to\infty} \frac{P(x)}{2^x}=0$. 
A: The factorial can't be expressed as a polynomial. Polynomials as really quite simple objects and if you could write the factorial as a polynomial then you would most certainly have heard about it.
On really cool thing that you can do though is to extend the factorial to a function on the complex plane. This function, called the Gamma Function, has the property that $\Gamma(n)=(n-1)!$ for all positive integers $n$. However, $\Gamma$ is defined on all complex numbers except the non positive integers.
$$\Gamma(z) := \int_0^{\infty}x^{z-1}\mathrm{e}^{-x}~\mathrm{d}x$$
