Fundamental Theorem of Algebra

Date: 01/25/2001 at 11:45:27
From: Nataria Joseph
Subject: The Fundamental Theorem of Algebra

What exactly is the Fundamental Theorem of Algebra?

Date: 01/25/2001 at 11:58:49
From: Doctor Schwa
Subject: Re: The Fundamental Theorem of Algebra

The fundamental theorem of algebra  says that any polynomial,

ax^n + bx^(n-1) + ...

with the a, b, etc. coefficients real or complex,
can be factored completely into

(x - r)(x - s) ...

where the r, s, etc. are complex numbers.

That is, every polynomial has solutions in the complex numbers.

The interesting thing about that, to me, is that we are in some
sense done inventing numbers once we get up to the complex.

Equations like x + 5 = 2 make you invent negative numbers;
equations like 2x = 3 make you invent fractions;
equations like x^2 = 2 make you invent irrationals;
equations like x^2 = -1 make you invent imaginaries;
but even equations like x^2 = i can be solved just using more
imaginary (complex) numbers.  You don't need anything new!

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/