# Product of Rational Polynomials

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## 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:

$\map f X = \cont f \cdot \map {f^*} X$
$\map g X = \cont g \cdot \map {g^*} X$

where:

$\cont f, \cont g$ 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:

$\map {h^*} X = \map {f^*} X \map {g^*} X$

## Proof

$\cont h \cdot \map {h^*} X = \cont f \cont g \cdot \map {f^*} X \map {g^*} X$

and this expression is unique.

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

$\cont f \cont g = \cont f \cont g > 0$.

The result follows.

$\blacksquare$