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138 4. Equations

4.4 The Fundamental Theorem of Algebra:

Intersecting Curves

The Fundamental Theorem of Algebra states that every polynomial of posi-
tive degree over C has at least one complex zero. Since there is no algorithm
which will allow us to construct such a root in general, the proof of this
theorem has to be tackled indirectly in a way which exploits the structure
of the complex plane. We look at the complex numbers geometrically. Each
complex number z = x + yi can be represented by a point (x, y) in the
Cartesian plane. A polynomial equation in the variable z is equivalent to
a pair of real equations in the variables x and y, whose loci are curves in
the plane. The intersections of these curves correspond to solutions of the
polynomial equation. Thus, the proof of the fundamental theorem depends
on ensuring that certain curves intersect. The exercises in this section will
examine the situation when the polynomial has low degree and suggest how
one proceeds with the task in general.

Exercises

1. Let a, b be real numbers with a # 0. Show that, if z = x + yi, with x
   and y real, the complex equation

az+b=O

is equivalent to the simultaneous real system

ax+b=O ay = 0

in the sense that any solution of one corresponds to a solution to

the other. Solve the real system graphically and argue that there is

always a’unique solution.

2. Illustrate graphically the solutions in the complex plane of the fol-
   lowing equations:

(a) 3% + 4 = 0

(b) (2 + i)z + (-3 + 4i) = 0.

3. Let a = p + qi and b = r + si be two complex numbers with a # 0.
   Find a real system of two simultaneous equations equivalent to the
   complex equation az + b = 0. Solve the system graphically and argue
   that it always has a unique solution.


4. (a) Let a, b, c be real numbers, with (I # 0. Show that the substitu-
   tion z = x + yi permits a reformulation of the complex equation

az2 + bz + c = 0