If a polynomial $f(x)$ in $\mathbb{C}[x]$ is divisible by $(x-a)^m$ but not by $(x-a)^{m+1}$ ($a$ is some complex number, $m \in \mathbb{Z}_+$ ), we say that $x = a$ is a zero of the polynomial with multiplicity $m$ or alternatively a zero of order $m$ .

Generalization of the multiplicity to real and complex functions (by rspuzio): If the function $f$ is continuous on some open set $D$ and $f(a) = 0$ for some $a \in D$ , then the zero of $f$ at $a$ is said to be of multiplicity $m$ if $\frac{f(z)}{(z\!-\!a)^m}$ is continuous in $D$ but $\frac{f(z)}{(z-a)^{m+1}}$ is not.

If $m \geqq 2$ , we speak of a multiple zero; if $m = 1$ , we speak of a simple zero. If $m = 0$ , then actually the number $a$ is not a zero of $f(x)$ , i.e. $f(a) \neq 0$ .

Some properties (from which 2, 3 and 4 concern only polynomials):

1. The zero $a$ of a polynomial $f(x)$ with multiplicity $m$ is a zero of the derivative $f'(x)$ with multiplicity $m\!-\!1$ .
2. The zeros of the polynomial $\gcd(f(x), f'(x))$ are same as the multiple zeros of $f(x)$ .
3. The quotient $\displaystyle\frac{f(x)}{\gcd(f(x), f'(x))}$ has the same zeros as $f(x)$ but they all are simple.
4. The zeros of any irreducible polynomial are simple.