Content of Rational Polynomial is Multiplicative/Proof 1

Proof
Let $f^* = \dfrac 1 {\cont f} f$, $g^* = \dfrac 1 {\cont g} g$

By Content of Scalar Multiple:
 * $\cont {f^*} = \cont {g^*} = 1$

That is, $f^*$ and $g^*$ are primitive.

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

Then: