Solving univariate polynomial equations is one of the oldest problems in mathematics. One approach to finding all roots of polynomial of degree n is to find better and better approximations for all roots simultaneously. I will present two methods using this approach, the Weierstrass-Durand-Kerner method and the Aberth-Ehrlich method. Both can be interpreted as higher-dimensional complex dynamical systems, and I will discuss some findings about their low-period cycles for cubic polynomials.