Computing with the exact root of a polynomial: rootof

### 5.27.21 Computing with the exact root of a polynomial: rootof

The rootof command finds the value of one polynomial at a root of another.

- rootof takes two arguments:
  P and Q, two polynomials given by the lists of their coefficients.
- 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:

  −i | ⎛ ⎜ ⎝ | − | √ |
  2
  | + | √ |
  3
  | ⎞ ⎟ ⎠

- 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:

  −2 | √ |
  6
  | −5