I am exploring root-finding algorithms and methods for finding roots of third degree polynomials. For the bisection method, I understand that one must specify an interval which contains the roots if $f(a)f(b) = 0$. finding the root would be easily achieved with a cubic equation with one real root. However, in the case of two or more real roots, the method will converge towards only one, depending on location.
In order to combat this problem, I was thinking of a method that allows us to divide the polynomial into intervals such that each interval only contains one root. is there a way to do that, and are there alternatives such that I do not use other method than the bisection method?
Additionally, I read that Sturm's theorem can define an interval in which a root is located, but how effective is it for multiple roots, and does the condition of no square apply to the polynomial itself or its factors?
thanks in advance