Compute with the exact root of a polynomial : rootof

### 2.24.18 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)` denotes 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)`