If a polynomialMathworldPlanetmathPlanetmathPlanetmath f(x) in [x] is divisible by (x-a)m but not by (x-a)m+1 (a is some complex numberMathworldPlanetmathPlanetmath,  m+), we say that   x=a   is a  zero of the polynomial with multiplicity m or alternatively a zero of order m.

GeneralizationPlanetmathPlanetmath of the multiplicity to real ( and complex functions (by rspuzio):  If the function f is continuousMathworldPlanetmathPlanetmath on some open set D and  f(a)=0  for some  aD, then the zero of f at a is said to be of multiplicity m if f(z)(z-a)m is continuous in D but f(z)(z-a)m+1 is not.

If  m2, 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)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(x) with multiplicity m-1.

  2. 2.

    The zeros of the polynomial gcd(f(x),f(x)) are same as the multiple zeros of f(x).

  3. 3.

    The quotient f(x)gcd(f(x),f(x)) has the same zeros as f(x) but they all are .

  4. 4.

    The zeros of any irreducible polynomialMathworldPlanetmath are .

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