Looking For A Theorem About n-th Root There is a theorem which I do not fully remember that goes something like: If $x_0$ is a root of $(x-x_0)^n$ than if can be written as $(x-x_0)^m\cdot q(x)$ 
Which theorem is it?

This could be the Factor Theorem:

Let $x_0$ be a root of the polynomial $q(x)$, of degree $n$. Then $q(x)$ may be written as $(x-x_0)^m r(x)$ for some $m$ and some polynomial $r$, of degree $n-m$.

Or the Fundamental Theorem of Algebra:

Every non-constant polynomial has a root in $\mathbb{C}$.

It looks a bit like the definition of the order of a root of a function:

Let $f$ be holomorphic at $a$, with a root at $a$, and not identically zero, and write it as $\sum_{n=1}^{\infty} c_n (z-a)^n$. There is a least $N$ such that $c_N \not = 0$; then $f(z) = (z-a)^N g(z)$ for some power series $g$ where $g(a) \not = 0$. We say $N$ is the order of the zero of $f$ at $a$.

A corollary of the uniqueness of analytic continuation, which lets us remove poles up to a polynomial factor:

Suppose $f$ is holomorphic on $U \setminus \{z_0\}$, and suppose $|f| \to \infty$ as $z \to z_0$. Then there is a unique $k \in \mathbb{N}$ and unique holomorphic $g: U \to \mathbb{C}$ such that $g(z_0) \not = 0$ and $(z-z_0)^k f(z) = g(z)$ for all $z \in U \setminus \{z_0\}$.

