2

Let $n > 0$ and let $a_{n-1}, ..., a_0$ be integers. Prove that, if $x^n + a_{n-1}x^{n-1} + ... + a_0 = 0$ has a rational root, then such root is an integer.

I understand that posting questions without showing your work is frowned upon, but I don't know where to start with this one.. thanks in advance.

Sigma
  • 1,836
  • 15
  • 35

1 Answers1

4

Denote the rational root by $\frac{p}{q}$ where $p\in\mathbb{Z},q\in\mathbb{N}$ and $\gcd(p,q)=1$. We have:

$\frac{p^n}{q^n}+a_{n-1}\frac{p^{n-1}}{q^{n-1}}+...+a_0=0$

Multiply the equation by $q^n$. Then we get:

$p^n+qa_{n-1}p^{n-1}+q^2a_{n-2}p^{n-2}+...+q^na_0=0$

This implies that $q$ divides $p^n$. But since $\gcd(p,q)=1$ we also have $\gcd(p^n,q)=1$, and hence if $q$ divides $p^n$ then $q=1$.

Mark
  • 39,605
  • It's interesting how a proof becomes "obvious" after you learn it.. I was staring at this problem for some 15 minutes and came up with nothing. – Sigma Sep 07 '19 at 21:35