Content of Rational Polynomial is Multiplicative

Theorem
Let $h \in \Q \sqbrk X$ be a polynomial with rational coefficients.

Let $c_h = \cont h$ denote the content of $h$.

Then for any polynomials $f, g \in \Q \sqbrk X$ with rational coefficients:
 * $\cont {f g} = \cont f \cont g$

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

By Content of Scalar Multiple:
 * $c_{\tilde f} = c_{\tilde g} = 1$

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

By Gauss's Lemma on Primitive Rational Polynomials, it follows that $\tilde f \tilde g$ is primitive.

Then: