It is clear that when you have two polynomials f and g, there exist uniquely determined polynomials q and r such as $f=qg+r$ (1), where $\deg{r}\leq \deg{g}$.
But how do you extend this theorem for polynomials such as $f(x,y)$ which are being divided to $g(x,y)$ and also what if the two poynomials were $f(x,y,z)$ and $g(x,y,z)$.
This is what I mean: For example, let $f(x,y,z)=x^3+y^3+y^3-x^2-y^2-z^2$ and $g(x,y,z)=xy+yz+zx$. If you divide $f$ to $g$ then would there hold a relation similar to (1)? What if the polynomials were arbitrary?
After writing the general statement, could you also provide an example for some specific polynomials, please? Thank you in advance!
