Root Locus of Some High-order Polynomials

For a third-order complex polynomial with its roots bounded by a circle, Marden’s theorem provides a useful way to evaluate the locus of its critical points. We construct a fourth-order complex polynomial by restricting its roots to ellipse and hyperbola. The locus of its critical points can be expressed by a formula. In addition, if the roots of complex polynomials are not restricted, their critical points will not occupy some regions.