Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 4 / 2E / a / Solution

Write the rational number as $r=p/q$ with coprime integers $p,q$ and $q>0$. Substituting into the monic polynomial and multiplying by $q^n$ givesAll terms except the first are divisible by $q$, so $q\mid p^n$. If $q>1$, it has a prime factor dividing $p^n$, and the Euclid lemma implies that this prime divides $p$, contradicting coprimality. Thus $q=1$, proving this special case of the rational root theorem: