1.13 General remarks

Although we didn’t prove this, it is true that if $\hbox{degree}\,f<\hbox{degree}\,g$ then the polynomial term is zero. Hence the first step is always to divide $f(x)$ by $g(x)$ as in frame 1.3, separating off the polynomial term. The rational function which remains will then be a sum of terms of the form $\frac{\gamma}{(x-a)^{r}}$ and $\frac{\alpha+\beta x}{Q(x)^{r}}$.

Warning 1: When $g(x)$ has an irreducible quadratic factor, it is not enough to consider expressions of the form $\frac{B}{Q(x)^{r}}$ with $B$ a constant.

Warning 2: When $g(x)$ has an irreducible quadratic factor or a squared linear factor, the usual shortcut for finding the coefficients via ‘evaluating at roots’ is not available.