Cubic and Quartic Formulae

[parisse]

If you are showing exact formula for 3rd or 4th order equations to your students, you should really emphasize that they should in fact never be used. Numerically, it's much more precise to run iteratives methods to find root approximations. Symbolically, it's way more efficient to use algebraic extensions of Q (rootof in Xcas, unfortunately not available to the user on the Prime, not my choice...).
Xcas has a small user program (comments are in French) to demonstrate Cardan formula that should be easy to adapt to the Prime

Code:


cardan(P,x):={
  local b,p,q,d,V,u,v,x1,x2,x3,n,j;
  j:=exp(2*i*pi/3);
  V:=symb2poly(P,x);
  n:=size(V);
  if (n!=4){
    throw(afficher(P)+" n'est pas de degre 3");
  }
  // Reduction de l'equation
  V:=V/V[0];
  b:=V[1];
  V:=ptayl(V,-b/3);
  p:=V[2];
  q:=V[3];
  // on est ramen￩ ￠ x^3+p*x+q=0
  // x=u+v -> u^3+v^3+(3uv+p)(u+v)+q=0
  // On pose uv=-p/3 donc u^3+v^3=-q et u^3 et v^3 sont solutions
  // de u^3 v^3 = -p^3/27 et u^3+v^3=-q
  // donc de x^2+q*x-p^3/27=0
  d:=q^2/4+p^3/27;
  if (d==0){
    // racine double
    return solve(P,x); 
  }
  d:=sqrt(d);
  u:=(-q/2+d)^(1/3);
  v:=-p/3/u;
  x1:=u+v-b/3;
  x2:=u*j+v*conj(j)-b/3;
  x3:=u*conj(j)+v*j-b/3;
  return [x1,x2,x3];
}:;

BTW, there is a Galois Field command on the Prime. Try GF(2,8) for example, then you can work in the non-prime field with 256 elements.