Suppose that $p(x)$ and $q(x)$ are two polynomials of the same order $n$, say $n=2$, and their coefficients are real numbers, but not zeros. In which case the following function $$q(x)=\frac{p(x)}{g(x)}$$
has a maximum or a minimum, where $g(x)\neq 0$, and does it matter that they are of the same order? Namely, in this case where the polynomials have the same order, is there any chance that they preserve monotonicity and as a consequence they do not have any optimum value? Is there any theory that covers these topics?