Factoring 3 Dimensional Polynomials? How do you factor a system of polynomials into their roots the way one can factor a single dimensional polynomial into its roots.
Example
$$x^2 + y^2 = 14$$
$$xy = 1$$
We note that we can find the 4 solutions via quadratic formula and substitution such that the solutions can be separated into $2$ groups of $2$ such that each group lies on either $(x + y + 4)$ and $(x + y - 4)$. Note that:
$$(x + y + 4)(x + y - 4) = 0$$
$$xy = 1$$
Is also an equivalent system.
How do I factor the bottom half?
Ideally if $g$ is a linear expression then my system should be
$$g_1 * g_2 = 0$$
$$g_3 * g_4 = 0$$
Such that the solutions to
Any of the subsystems of this system is a solution to system itself (note there are $4$ viable subsystems).
Help?

If you have equations of the form $p(x, y) = 0$ where $p$ is a polynomial, factoring that one is certainly worthwhile, It will give you more equations, but simpler ones. So, if you have:
$$
\begin{align*}
(x + y) (x^2 - y^2) &= 0 \\
(x - y) (x^2 + y^2) &= 0
\end{align*}
$$
you end up with $2 \times 2 = 4$ (pick one from each) systems of equations.
But if factoring polynomials in one variable isn't a walk in the park, with more variables it just gets worse.
