Content of Rational Polynomial is Multiplicative

Theorem
If $h \in \Q\left[X\right]$ is a polynomial with rational coefficients, let $c_h = \operatorname{cont}\left({h}\right)$ denote the content of $h$.

Then for any polynomials $f,g \in \Q\left[{X}\right]$ with rational coefficients we have:
 * $\displaystyle \operatorname{cont}\left({fg}\right) = \operatorname{cont}\left({f}\right)\operatorname{cont}\left({g}\right)$

Proof
Let $\tilde f = c_f^{-1} f$, $\tilde g = c_g^{-1} g$

By Content of Scalar Multiple, we have $c_{\tilde f} = c_{\tilde g} = 1$.

That is, $\tilde f$ and $\tilde g$ are primitive.

By the Gauss' lemma, it follows that $\tilde f \tilde g$ is primitive.

Now,