Let's say that I have one polynomial $a(x)$ of degree $n$ with coefficients over $\mathbb{F}_p$, where $\mathbb{F}_p$ is a finite field of size $p$.
Asuming that $(x - r) \mid a(x)$ (i.e., that $r \in \mathbb{F}_p$ is a root of $a(x)$) then we have that the quotient polynomial: $$q(x) := \frac{a(x)}{x-r}$$ is a polynomial of degree $n-1$.
Let $S$ be a subset of $\mathbb{F}_p$ of size "much bigger" than n (for instance, imagine that $|S| = 2n$) that does not includes the root $r$. What can we say about the degree of the interpolation of $q(x)$ over $S \subseteq \mathbb{F}_p$ when $(x-r) \nmid a(x)$?
Clearly, $q(x)$ is not a polynomial but a rational function. However, there is nothing that does not allow me to evaluate $q(x)$ over $S$ and then interpolate the resulting evaluations over $S$. At this point, we can say that the resulting interpolation is a polynomial of degree lower than $|S|$.
Can we say better than this? What can we say about the degree of the interpolation compared to the degree of $a(x)$? Is the degree of the interpolation necessarily higher than $n$?
Intuitively, the interpolation has to have a degree bigger than $n$; since otherwise the division would be clear.