From: "Steven Sivek" <stevensivek@hotmail.com>
Subject: Re: Rational Zeros Theorem
Date: 6 Sep 1999 16:25:49 GMT
Newsgroups: sci.math

If you are referring to the theorem (I've heard it called the Rational Root
Theorem or something similar) that states that for all rational solutions
p/q of a polynomial equation a_n*x^n + a_(n-1)*x^(n-1) + ... + a_0, p is a
factor of +/- a_0 and q is a factor of +/- a_n, the proof is as follows:

Let x=p/q, where p/q is a rational root of the equation and (p,q)=0.  The
equation becomes a_n*(p/q)^n + ... + a_0 = 0.
Multiplying by q^n gives (a_n)(p^n) + (a_(n-1))(p^(n-1))(q) + ... +
(a_0)(q^n) = 0.

Examining the equation mod p, all terms except the last one have factors of
p and therefore cancel, leaving a_0*(q^n) = 0 (mod p).  Since p and q are
relatively prime, dividing by q^n gives a_0 = 0 (mod p) and p is therefore
a factor of +/- a_0.

If we look at the equation mod q instead, all terms except the first cancel
and we have (a_n)(p^n) = 0 (mod q).  Since p and q are relatively prime, we
divide by p^n to get a_n = 0 (mod q) and q is therefore a factor of +/-
a_n.  QED.

You can also use this to state instead that (a_0/a_n) is a rational
multiple of p/q.

Steven Sivek
stevensivek@hotmail.com

> Does anyone know where I could find a proof of the "Rational Zeros Theorem"?
> 
> Thanks,
> 
> Jonathan Nambia