I'd have to look at Cardano's book before I could answer that question certainly, but the complex intermediate steps are, as far as I know, unavoidable. I think the library has a copy of Ars magna; I'll look it up next time I'm on campus.
If the three roots u,v,w are real and the x^2 term is zero, they must satisfy u+v+w=0, so either two of them are negative and one positive or two are positive and one negative. If we write it in the form x^3+px+q=0, p is uv+uw+vw and q is -uvw; in either case p and q wind up with the same sign, so (forcing coefficients to be positive) it's either x^3+mx+n=0 or x^3=mx+n.
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Date: 2007-02-03 08:44 pm (UTC)If the three roots u,v,w are real and the x^2 term is zero, they must satisfy u+v+w=0, so either two of them are negative and one positive or two are positive and one negative. If we write it in the form x^3+px+q=0, p is uv+uw+vw and q is -uvw; in either case p and q wind up with the same sign, so (forcing coefficients to be positive) it's either x^3+mx+n=0 or x^3=mx+n.