> I was curious if the Cardan formulas are used in the computation of roots of cubic and quartic polynomials.
Yes.
> I could not find them.
Try solve(a*x^4+...+e,x).
> In general, one would like to find symbolic solutions of rational
> polynomials when possible
Maxima does this for all polynomials up to quartics, and for some
higher-degree polynomials, notably some simple cases (x^a+b), those
which factor over the rationals into quartics or less, and those which
are compositions of quartics or less. Try, for example
solve(x^6-x^4-x^3+2*x-2,x)
(cf. factor)
solve(x^9-6*x^6+12*x^3-10,x)
(cf. polydecomp)
In general, though, the symbolic solutions are a huge mess, and not
much use to anyone.... I suppose there are some cases where they
could be useful, e.g.
sols: solve(x^4+a*x-1,x)$
taylor(part(sols,1,2),a,0,3);
(look at solutions where a<<1)
Maxima does not currently handle other cases of quintics+.
-s