Rephrase: Prove for any rational root of a polynomial ... that the numerator $p$ is a factor of $a_0$ and the denominator $q$ is a factor of $a_n$.

Question

Prove for any rational root of a polynomial with integer coefficients, $$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 \space\space\space\space (a_n \neq 0)$$ if written in lowest terms as $p/q$, that the numerator $p$ is a factor of $a_0$ and the denominator $q$ is a factor of $a_n$. (This criterion permits us to obtain all rational real roots and hence to demonstrate the irrationality of any other real roots.)

I'm not precisely sure what this exercise is requesting. Can someone rephrase this for me?

$^\text{If using the `translation-request` tag is inappropriate, feel free to change it. I'm not aware of other tags regarding rephrasing an exercise.}$

Answer 1

For any rational root of the polynomial of the form ${p\over q}$ where $\gcd(p,q)=1$, p is a factor of $a_0$ and $q$ is a factor of $a_n$.