On the Fundamental Theorem of Algebra

We begin our story with a 2-dimensional plot of a polynomial c urve in one variable, y = f (x). Our choice off (x), above left, is the cubic expression x3− x2−4x−6. It has a single real root, x = 3: a value for whichf (x) = 0, with the curve meeting the horizontal axis. But we can picture the same curve living in t hree dimensions, as shown at (a), with a second horizontal axis at right angles to the real number line. This represents all real number multiples of i, the imaginary number whose square is −1. Together, the two axes define a plane, called the complex pl ane. We may write f (z) instead off (x) to indicate that our variable is now a complex number, z = a + ib. Across the complex plane, f takes complex values: to plot them would require two more dimensions! Instead we plot the real and ima g nary parts off (z) separately. At (b) we see that, at z = −1± i, the surface plotting the real part of y = f (z) passes up through the complex plane: potentially we have tw o more roots! And indeed a plot of the imaginary part of f (z), shown at (c), confirms that the line x = −1 is indeed crossed by f (z) at±i. Function f (z) = z3 − z2 − 4z − 6