Exploring the Fundamental Theorem of Algebra

What do we see? To the left is a circle centred at the origin with radius where A lies on the x-axis. The point, z, lies on this circle. We think of z as a complex number. Indeed, z can be any complex number. Here . To the right there appears to be another circle, but this is not so. The closed path is the locus of where p is some polynomial with complex coefficients. Here . Check the following: Rotating z through one revolution about the circle causes w to rotate one revolution about its locus. As the radius of the circle diminishes (), the locus of w approximates a circle of radius r centred at 1. As this radius increases, the locus of w behaves strangely! Try A = 0.4, 0.6, 0.8 and 1. Describe what you see.