Question about a comment on a different question. A rational root of a monic polynomial over $\Bbb Z[x]$ is an integer

Question

This questions contains a comment that states that $\gcd(r,s)\ne 1$ (which leads to a contradiction) where our rational root is $r/s$.

Why would that be true?

Answer 1

Multiplying by $s^n$ gives $$r^n + a_{n-1}r^{n-1}s + \dots+ a_1rs^{n-1} + a_0s^n = 0$$ and so $$r^n=-s(a_{n-1}r^{n-1}+\dots+a_1rs^{n-2}+a_0s^{n-1}).$$ Thus $s\mid r^n$, which implies any prime factor of $s$ is also a prime factor of $r$. So, we conclude $\gcd(s,r)\neq 1$ unless $s$ has no prime factors.