Solution of the Equations

FUNDAMENTAL THEOREM OF ALGEBRA

Theorem. Suppose p(x) is any polynomial of degree n > 0. Then there is at
least one complex number which is a solution of the equation p(x) = 0.

Explanation: Suppose Then,
as before, for large x-values, p(x) behaves much like its highest degree term:
p(x) ≈ a_nxⁿ. For convenience, let us assume a_n = 1. (As far as looking for
roots is concerned, it makes no difference. If a_n = 17, for example, we could get
a new polynomial with the same roots by dividing every coefficient by 17.)

Let r be a large enough radius to make p(x) ≈ xⁿ a good approximation (whatever
that means). Then p, as well, should send the disk D of radius r around
0 in C to the disk D' of radius rⁿ, wrapping the boundary of D n times around
the boundary of D'.

Now we apply our two-dimensional counterpart to the IVT: Since y = 0 is contained
in the disk D', there must exist at least one point x in D so that p(x) = 0.
In other words, the polynomial has a root!

yay!

The area of mathematics which seeks to understand continuity and connectedness
for higher-dimensional shapes is called Topology. Some of topologists’ tools and
big ideas include:

• Far-reaching generalizations of theorems like our “Two-dimensional Intermediate
Value Theorem”.

• Precise notions of what it means for a curve to “wrap n times around
another,” in terms of winding numbers.

• Generalizations of winding numbers for wrapping higher-dimensional shapes
around higher-dimensional holes, using homology groups.

• Knot theory.

• Bottles whose insides and outsides are indistinguishable.

. . . to name a few. It has been said that topologists can’t tell their donuts from
their coffee mugs. But in view of all their other redeeming qualities, that has
little practical significance.

You might think, then, that algebraists would have a tough time coping with
the fact that their so-called “Fundamental Theorem” isn’t really a pure algebra
theorem at all!

Well, yes. People can be cruel.

But this fact really just serves to illustrate that no field of mathematics is entirely
self-contained. And even in this case, algebra can take over from here to provide
a much improved, “prescription strength” form of the Fundamental Theorem.

Consider the polynomial p(x) = x² − 6x + 9. Then p(x) factors as (x − 3)²,
so even though x = 3 is the only solution to the equation p(x) = 0, there is a
natural sense in which it should count twice. More generally, we have

FUNDAMENTAL THEOREM OF ALGEBRA, strong form

Theorem. Suppose p(x) is any polynomial of degree n > 0. Then, if we
count repeated roots as many times as they occur, the equation p(x) = 0 has
exactly n roots in the complex numbers.

Explanation: If p(x) has degree n > 0, the regular-strength theorem shows
that at least one root exists. Pick one, and call it c₁. Since c₁ is a root, (x−c₁)
must be an exact divisor of p(x). This means that is another polynomial.
Call it p₁(x), and notice that its degree will be one less than the degree of p.

Now, if n were 1, then p₁(x) would have degree 0, and hence be a constant. This
constant polynomial would have no more roots, so c₁ would be the one and only
root of p.

Otherwise, p₁(x) has degree ≥ 1, and the regular-strength theorem applies again,
to give another root, c₂. But then is another polynomial. Call it p₂.

Repeating this process exactly n times, we finally get to the stage where p_n(x)
has degree zero, and we have found exactly n roots,

Double-yay!

Here is another way we can interpret the Fundamental Theorem of Algebra: The
German mathematician Leopold Kronecker said, quotably,

God created the natural numbers, and all the rest is the work of
man.

In fact, we can think of the logical development of number systems as a gradual
expansion through time, to give solutions to more and more equations and
increase versatility.

• In the natural numbers, we can always add any pair of numbers. But
subtraction only works sometimes. To solve equations like x + 3 = 2, we
need to understand what the difference 2−3 is, so we expand N to include
negative whole numbers, and get the set of Integers:

Z = {. . . − 3,−2,−1, 0, 1, 2, . . .}.

• In the integers, we can always multiply, but only sometimes divide. To
solve equations like 3x = 5, we need to make sense of 5/3, so we expand
the number system again to include fractions, and get the set of Rational
Numbers:

• But the rational numbers don’t include numbers like π and and also
(as we have seen) fail to adequately reflect our intuitions about continuity
and linear connectedness. Expanding to R solves those problems.

• And in R, equations like x² = −1 still have no solutions. So we introduce
another number, i, with the property that i² = −1, and allow it to be
added and multiplied by each number in R. This gives

The Fundamental Theorem of Algebra then says:

We’re done.

We now have as many solutions as we need to any sort of polynomial equation
we can imagine, and we won’t need to expand any more.

So, as you can see, the algebraists really have nothing to be ashamed of. But
everyone can use a little encouragement now and then. So if you see one...

Hug an Algebraist Today!