I've been playing around with Cardano's formula, and I've come away a little bit disappointed : I'm under the impression that for a polynomial with rational coefficients, the formulas yield an intractable mess unless either at least one root is rational (in which case one can just use the rational root theorem) or at least one root is some variation on cos(pi/9) or cos(pi/7), essentially because these numbers have degree 3 minimal polynomials : https://en.wikipedia.org/wiki/Minimal_polynomial_of_2cos(2pi/n)

Does anyone have any cubics that are solvable by hand and yield "pretty" roots ?

I've been playing around with Cardano's formula, and I've come away a little bit disappointed : I'm under the impression that for a polynomial with rational coefficients, the formulas yield an intractable mess unless either at least one root is rational (in which case one can just use the rational root theorem) or at least one root is some variation on $\cos(\pi/9)$ or $\cos(\pi/7)$, essentially because these numbers have degree 3 minimal polynomials. See Wikipedia's "Minimal polynomial of $2\cos(2\pi/n)$" entry.

Does anyone have any cubics that are solvable by hand and yield "pretty" roots ?