Given a depressed cubic equation $$x^3+px+q=0,$$ with $p$ and $q$ negative real numbers, is there an easy upper bound for its real roots in terms of $p$ and $q$? The sharper the better, of course.
EDIT: I had little clue how to approach this problem. I tried substituting polynomials of low degree in $p$ and $q$ for $x$, such as $x=p+q$ and $x=\sqrt{-p}$ to see whether the result would be positive or negative in general, but to little avail. I don't see how to approach this constructively, or how to find simple expressions to plug in that will always yield a positive result.
There is a general bound $M$ on the moduli of the (real or complex) roots $\,\xi_i\,$ in $\mathbf C$ of a polynomial of any degree $n$ in $\mathbf C[x]$, $ a_n x^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0\enspace (a_n\ne 0)$: $$|\xi_i|\le M=\max\biggl(1, \frac{|a_{n-1}|}{|a_n|}+\dots+\frac{|a_{0}|}{|a_n|}\biggr). $$ For a depressed cubic, this yields the bound $\; M=\max(1,|p|+|q|)$.
On the other hand, Sturm's algorithm yields the exact number of distinct real roots in a given interval.