I'm not sure how to find the degree of the ratio $\frac{p(x)}{q(x)}$ of polynomials $p(x)$ and $q(x)$ when $\deg(q(x)) > \deg(p(x))$. I couldn't find an answer. For example, $\deg(p(x))=10$ and $\deg(q(x))=26$. Intuitively the degree of the ratio should be $-\infty$, and the degree of the remainder should be $10$.
Want to be sure if I'm right. Any help appreciated, thanks!
If I have understood you correctly I think you are right.
In the case when the divisor has greater degree than the one we are diving, the Division Algorithm says that $p(X)=0.q(X)+ p(X)$, and so the quotient is $0$ and the remainder is $p(X)$.