Show that any polynomial $p(x) \in \mathbb{Q}[x]$ can be written as $p(x) = tq(x)$ where $t \in \mathbb{Q}$ and $q(x) \in \mathbb{Z}[x]$ is primitive.
I started my proof by defining $p(x)$ as $(\frac{q}{r})_n x^n + \dots + (\frac{q}{r})_0$. Then I defined $t \in \mathbb{Q}$ as the product of greatest common factor of the coefficients of $p(x)$. I don't think this will work. How can I guarantee that after dividing $p(x)$ by $t$ I will get a primitive polynomial?