Rational Functions

A rational expression in $x$ is the ratio $\dfrac{P(x)}{Q(x)}$ of two polynomials with $Q(x)\ne0$, defined for every $x$ where $Q(x)\ne0$. If $\deg P\ge\deg Q$, dividing (§2.6.1's division algorithm) gives

$\dfrac{P(x)}{Q(x)}=f(x)+\dfrac{r(x)}{Q(x)},\qquad \deg r<\deg Q,$ …