Product of Rational Polynomials

Theorem
Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$.

Let $\map f X, \map g X \in \Q \sqbrk X$.

Using Rational Polynomial is Content Times Primitive Polynomial, let these be expressed as:
 * $\map f x = c_f \cdot \map {f^*} X$


 * $\map g X = c_g \cdot \map {g^*} X$

where:
 * $c_f, c_g$ are the content of $f$ and $g$ respectively
 * $f^*, g^*$ are primitive.

Let $\map h X = \map f X \, \map g X$ be the product of $f$ and $g$.

Then:
 * $c_h = c_f c_g$
 * $\map {h^*} X = \map {f^*} X \, \map {g^*} X$

Proof
We have, by applications of Rational Polynomial is Content Times Primitive Polynomial:


 * $c_h \cdot \map {h^*} X = c_f c_g \cdot \map {f^*} X \, \map {g^*} X$

By Gauss's Lemma on Primitive Polynomials we have that $\map {f^*} X \, \map {g^*} X$ is primitive.

As $c_f > 0$ and $c_g > 0$, then so is $c_f c_g > 0$.

By the uniqueness clause in Rational Polynomial is Content Times Primitive Polynomial, the result follows.