Perturbing the Constant Coefficient of a Complex Polynomial

This Demonstration plots the complex polynomial . For , the multiplicity of a root determines the number of cycles (cyan, blue, yellow, red) around the root (shown as a black dot). An example is shown in the first snapshot. You can vary , , , , , and . Small changes in the constant term, which makes , transforms the polynomial to a form , with all of the roots of the new polynomial being simple.