Asymptotes and Partial Fractions
Use polynomial division and decomposition to simplify rational expressions.
When the numerator degree is greater than or equal to the denominator degree, polynomial division helps reveal the graph structure.
Division idea\[\frac{P(x)}{Q(x)}=\text{quotient}+\frac{\text{remainder}}{Q(x)}\]
Oblique asymptote
If division gives \(f(x)=mx+c+\frac{r(x)}{Q(x)}\) and the fraction tends to zero, then \(y=mx+c\) is an oblique asymptote.
Partial fractions
A typical decomposition is
\[\frac{5x+1}{(x-2)(x+3)}=\frac{A}{x-2}+\frac{B}{x+3}\]This is especially useful before integration.