A question about polynomials.

Question

My book says

If a rational number $m/n$, for $(m,n)=1$, is a root of the polynomial $a_rx^r+a_{r-1}x^{r-1}+\dots+a_0$, where $a_0,a_1,\dots,a_r\in\Bbb{Z}$, then $n|a_r$ and $m|a_0$.

I was under the impression that such a polynomial with integer coefficients has only integral rational roots.

Is this statement just saying that $m|a_0$ and $1|a_r$?

Thanks in advance!

Answer 1

It is easy to see that the integer-coefficient polynomial $2x-1$ does not have integer roots.

What you're thinking of requires the polynomial to be monic, i.e. have leading coefficient $a_r=1$. In the case when $a_r=1$, the statement as given in your book becomes (in part) that $n\mid 1$, so that $n=\pm 1$, so that the rational root $\frac{m}{n}$ is an integer.