From: "Steven Sivek" 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