Definitions from Rational functions

A function $f$ is a rational function if it is of the form $f(x)=\frac{P(x)}{Q(x)}$, where $P$ and $Q$ are polynomials and $Q$ is not the zero polynomial. Its real domain excludes the zeros of $Q$.

A rational function $f$ is proper if $\deg P<\deg Q$. If $f$ is not proper, long division writes it as a polynomial plus a proper rational function,

with $\frac{R(x)}{Q(x)}$ proper.