multiplicity
If a polynomial^{} $f(x)$ in $\u2102[x]$ is divisible by ${(xa)}^{m}$ but not by ${(xa)}^{m+1}$ ($a$ 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 (http://planetmath.org/RealFunction) 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 $\frac{f(z)}{{(za)}^{m}}$ is continuous in $D$ but $\frac{f(z)}{{(za)}^{m+1}}$ 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. $f(a)\ne 0$.
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 ${f}^{\prime}(x)$ with multiplicity $m1$.

2.
The zeros of the polynomial $\mathrm{gcd}(f(x),{f}^{\prime}(x))$ are same as the multiple zeros of $f(x)$.

3.
The quotient $\frac{f(x)}{\mathrm{gcd}(f(x),{f}^{\prime}(x))}$ has the same zeros as $f(x)$ but they all are .

4.
The zeros of any irreducible polynomial^{} are .
Title  multiplicity 
Canonical name  Multiplicity 
Date of creation  20130322 14:24:18 
Last modified on  20130322 14:24:18 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  14 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 12D10 
Synonym  order of the zero 
Related topic  OrderOfVanishing 
Related topic  DerivativeOfPolynomial 
Defines  zero of order 
Defines  multiple zero 
Defines  simple zero 
Defines  simple 