So, if $\frac{a}{b}$ is a root for $p(x)$ then $b$ divides the constant term and $a$ divides the leading term. Since we are assuming that $\frac{a}{b}\in\mathbb{Z}$, then we know that $b=\pm1$ and since the leading coefficient is $1$ we know that $a=\pm1$ and so $\frac{a}{b}=\pm1$
(This isn't what you asked me to prove, but it's where my mind went and seems relevant)