# Exploring the Fundamental Theorem of Algebra

- Author:
- Maurice OReilly

- Topic:
- Algebra

What do we see? To the left is a circle centred at the origin with radius where . To the right there appears to be another circle, but this is not so. The closed path is the locus of where .
Check the following:
Rotating ), the locus of

*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*p*is some polynomial with complex coefficients. Here*z*through one revolution about the circle causes*w*to rotate one revolution about its locus. As the radius of the circle diminishes (*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.