Exploring the Fundamental Theorem of Algebra
- Maurice OReilly
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.
Now zoom out, so you can see the whole picture. Notice that the locus passes through the origin. Why is this? Now try A = 2, 3. Zoom out again. Describe what you see. Notice that as z rotates through a revolution, w does so twice. We say that the locus has winding number 2 (about the origin). Try A = 5, 10. Zoom out and describe. What about 50? Now explore all of this for with n = 3, 4, 5, ... (To do this, just right click on w and edit Object Properties.) Try any polynomial, p, of your choice, including ones with imaginary coefficients, and, in particular, with imaginary constant coefficients. No matter what polynomial, p, we choose, we see that for sufficiently small r, the locus of w is almost a (small) circle centred at , the constant coefficient in . On the other hand, for sufficiently large r, the locus is almost a (large) circle (with winding number n) centred at the origin. (What is the radius of each of these circles?) If , then the locus must, for some value of r, pass through the origin. Thus for some z, proving the Fundamental Theorem of Algebra. Of course, the word 'must' needs further scrutiny and relies on the fact that the locus of w varies continuously with r.