Roots of univariate polynomials via the euclidean algorithm

In this paper, we will consider univariate polynomials with real coefficients in degrees three, four, and five in great detail. Polynomial conditions will be found in terms of the coefficients of the univariate polynomial that will be able to describe the configuration of the roots. These conditions will be able to tell what the multiplicities of all the roots of the univariate polynomial are, as well as how many of these roots are real and how many are complex. In the scenario where the polynomial has real roots of different multiplicities, there will be conditions found that will describe the relative position of these real roots with respect to their multiplicities.