# multiplicity

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 (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 $\frac{f(z)}{(z\!-\!a)^{m}}$ 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. 1.

The zero $a$ of a polynomial $f(x)$ with multiplicity $m$ is a zero of the $f^{\prime}(x)$ with multiplicity $m\!-\!1$.

2. 2.

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

3. 3.

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

4. 4.
 Title multiplicity Canonical name Multiplicity Date of creation 2013-03-22 14:24:18 Last modified on 2013-03-22 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