Compute with the exact root of a polynomial : rootof

### 5.25.18 Compute with the exact root of a polynomial : rootof

Let P and Q be two polynomials given by the list of their coefficients then rootof(P,Q) gives the value P(α) where α is the root of Q with largest real part (and largest imaginary part in case of equality).
In exact computations, Xcas will rewrite rational evaluations of rootof as a unique rootof with degree(P)<degree(Q). If the resulting rootof is the solution of a second degree equation, it will be simplified.

Example
Let α be the root with largest imaginary part of Q(x)=x4+10x2+1 (all roots of Q have real part equal to 0).

- Compute 1/α. Input :
  normal(1/rootof([1,0],[1,0,10,0,1]))
  P(x)=x is represented by [1,0] and α by rootof([1,0],[1,0,10,0,1]).
  Output :
  rootof([[-1,0,-10,0],[1,0,10,0,1]])
  i.e. :

  1
  α
  | =−α3−10α

- Compute α2. Input :
  normal(rootof([1,0],[1,0,10,0,1])`^`2)
  or (since P(x)=x2 is represented by [1,0,0]) input
  normal(rootof([1,0,0],[1,0,10,0,1]))
  Output :
  -5-2*sqrt(6)