Proof of the Rational Root Theorem

Date: 11/13/2000 at 15:57:53
From: Jacob Bucksbaum
Subject: Proof of rational root theorem

I have been asked to prove the Rational Root Theorem, and I am lost. I 
was wondering if you could help?

Date: 11/13/2000 at 16:47:03
From: Doctor Rob
Subject: Re: Proof of rational root theorem

Thanks for writing to Ask Dr. Math, Jacob.

The theorem states that if a polynomial has a rational root, then the 
denominator of the root must divide the coefficient of the highest 
power term of the polynomial, and the numerator of the root must 
divide the constant term of the polynomial.

Start with a general polynomial equation with integer coefficients.

     a(n)*x^n + a(n-1)*x^(n-1) + ... + a(1)*x + a(0) = 0

Suppose it has a rational root r/s, where r and s are integers, and 
r/s is reduced to lowest terms (so r and s have no common factor). 
Substitute r/s for x in this equation.

First multiply this equation through by s^(n-1). You'll see that all 
but the first term are integers, and the first term is 

     a(n)*r^n/s  

That implies that a(n)*r^n/s must be an integer, so s divides evenly 
into a(n)*r^n. Since r and s have no common factor, it must be that s 
divides evenly into a(n).

Next multiply the equation through by s^n/r. You'll see that all but 
the last term are integers, and the last term is 

     a(0)*s^n/r

You finish.

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