  The isolation intervals of the real roots of the real symbolic monic cubic
polynomial $p(x) = x^3 + a x^2 + b x + c\,\,$ are found in terms of simple
functions of the coefficients of the polynomial (such as: $-a$, $-a/3$, $-c/b$,
$\pm \sqrt{-b}$, when $b$ is negative), and the roots of some auxiliary
quadratic equations whose coefficients are also simple functions of the
coefficients of the cubic. All possible cases are presented with clear and very
detailed diagrams. It is very easy to identify which of these diagrams is the
relevant one for any given cubic equation and to read from it the isolation
intervals of the real roots of the equation. A much-improved complete root
classification, addressing the signs (together with giving the isolation
intervals) of the individual roots, is also presented. No numerical
approximations or root finding techniques are used. Instead of considering the
discriminant of the cubic, criterion for the existence of a single real root or
three real roots is found as conditions on the coefficients of the cubic,
resulting from the roots of the auxiliary quadratic equations. It is also shown
that, if a cubic equation has three real roots, then these lie in an interval
$I$ such that $\sqrt{3}\sqrt{a^2/3 - b} \le I \le 2 \sqrt{a^2/3 - b}$,
independent of $c$. A detailed algorithm for applying the method for isolation
of the roots of the cubic is also given and it is illustrated through examples,
including the full mathematical analysis of the cubic equation associated with
the Rayleigh elastic waves and finding the isolation intervals of its real
roots.
