The work presented the solutions of polynomial algebraic equations by radicals and elementary operations in the coefficients with research based on books and articles; Sought to explore the various ideas developed in the demonstrations, discussions on the cases and ingenious artifacts involved, as well as some independent demonstrations; Were still treated, the special cases where the roots are subject to pre-established conditions and the coefficients obey a given law; We use Abel-Ruffini s theory and the implications of Galois s theory to justify the impossibility of a general solution by radicals of polynomials of degree greater than or equal 5 and the answer to this impasse with the emergence of numerical approximation methods. These theories and methods were treated in an elementary way, because they require other detailed work, which is beyond the scope of this work. Thus, we have seen algorithms that allow us to calculate, in cases from 1st to 4th degrees, the solutions of a polynomial algebraic equation in addition to special cases and numerical approximations. We use the algebraic computing and geometry programs: Maxima, Geogebra and Maple for approximations, drawings and graphs.