ROOTS WITH MULTIPLICITY

 

Author : Jean Michel Ferrard
Syntax : MZEROS ( expr , var )
Require : ******

It is said that is a root of the polynomial "P" if P()=0.

It is said that is root of P with the multiplicity m if (X-)m divides P, but not (X-)^(m+). One speaks about simple' root if m=1, double if m=2, triple if m=3, ... ... . One speaks of multiple root if m>1.
It is shown that the multiplicity m of a root "" of P is characterized by:

P()=p ' ()=p " ()=... =P^(m-) ()=0, and P(m)^()/=0 (the multiplicity appears as the order of the first derivative which is not cancelled at ).
The integrated functions "zeros()" and "czeros()" only give the different zero from a polynomial. The function "mzeros()" corrects this problem while returning the list with all the zeros.

Syntax is mzeros(P, x), where P is an polynomial expression (and not a list! compared to the variable "X" (x or another name of variable).

 

Exemple :

Here : P(x)=(x-1)^3*(x-)^2*(x+1) : roots (and multiplicities) are obvious.
One develops P and one seeks his roots with ZEROS(). One finds three the distinct zero without the multiplicities.
With mzeros(), one sees that 1 is a "triple root" and that is a "double root" of P.