Problem
ALG-B1-M09-P021 Rational root of a monic polynomial
#21
★★★★☆ Level 4 of 5
Prove that if a rational number \(\frac{p}{q}\) in lowest terms is a root of a monic polynomial with integer coefficients, then \(q=1\).
Substitute \(\frac{p}{q}\) and multiply by \(q^n\).
Let \(\left(\frac{p}{q}\right)^n+a_{n-1}\left(\frac{p}{q}\right)^{n-1}+\cdots+a_0=0\). Multiplying by \(q^n\), \(p^n+a_{n-1}p^{n-1}q+\cdots+a_0q^n=0\).
All terms except \(p^n\) are divisible by \(q\), so \(q\mid p^n\). Since \((p,q)=1\), we get \(q=1\).
This is the rigorous version of the integer-root rule.