### 5.33.1 Roots and poles of a rational function : froot

froot takes a rational function F(x) as argument.
froot returns a vector whose components are the roots and the poles of F[x], each one followed by its multiplicity.
If Xcas can not find the exact values of the roots or poles, it tries to find approximate values if F(x) has numeric coefficients.
Input :

froot((x`^`5-2*x`^`4+x`^`3)/(x-2))

Output :

[1,2,0,3,2,-1]

Hence, for F(x)=x5−2.x4+x3/x−2 :

- 1 is a root of multiplicity 2,
- 0 is a root of multiplicity 3,
- 2 is a pole of order 1.

Input :

froot((x`^`3-2*x`^`2+1)/(x-2))

Output :

[1,1,(1+sqrt(5))/2,1,(1-sqrt(5))/2,1,2,-1]

Remark : to have the complex roots and poles, check Complex in the cas configuration (red button giving the state line).
Input :

froot((x`^`2+1)/(x-2))

Output :

[-i,1,i,1,2,-1]