We will work with the last root to illustrate this. To make things a bit more convenient, we define the root as a function of the parameter. When the parameter Sigma is given a numerical value, Mathematica does the computations necessary to determine an approximate root that is precise enough to be isolated from the other roots of the polynomial. First, root approximations at machine precision are found using the Jenkins-Traub algorithm. The isolation is possible because, for a given root approximation, it is possible to find an enclosing circle in which an exact root of the polynomial is guaranteed to lie.
Isolation makes sure that none of the circles around the approximate values overlap.
Optionally, Mathematica allows you to use an exact root isolation algorithm that is more complicated and time consuming. Once the roots are isolated, they are ordered based on a canonical ordering. The pairs of roots switch to maintain the canonical ordering. Mathematica recognizes this, so the polynomial is given in its simplest form. Notice that the polynomial is recalculated to be the minimal polynomial with integer coefficients.