The Structure of the Real Roots of a Quintic Polynomial

For any polynomial , the discriminant of is a polynomial in the coefficients which is zero only when has multiple roots. However, the complete root structure of a general polynomial (i.e., the conditions that the coefficients must satisfy in order that the polynomial has a specified number of single, double, or multiple roots) is much harder to determine. Here we show graphically the solution of the problem for a polynomial of degree five of the form , where all the coefficients are real. We regard as coordinates of a point in the unit cube of length 10, with center at the origin. The constant coefficient , initially set to 0, can be varied by means of a slider taking values between -5 and 5. The parts of the cube corresponding to polynomials with a simple real root, three distinct real roots, five distinct real roots, and the discriminant are shown in different colors. The discriminant (the part of the cube corresponding to polynomials with multiple, possibly complex, roots) is also shown. The part of the discriminant colored red corresponds to polynomials with exactly four distinct real roots (one of them a double root); the rest of the discriminant is colored black. Note that the discriminant is not disjoint from the other areas shown, since a polynomial can have (for example) a simple real root and a double complex one, thus lying both on the discriminant and in the green area. To give the complete decomposition of the cube, the black part of the discriminant would have to be partitioned further, corresponding to different multiplicity structures of roots. This has been done algebraically (see the Details section) but is difficult to represent graphically so it is not done here.