Rational Fractions: Proper and Improper

A rational fraction is a quotient of two polynomials, $\dfrac{f(x)}{g(x)}$, with $g(x)\not\equiv0$. It is proper when the numerator's degree is less than the denominator's, and improper when the numerator's degree is greater than or equal to the denominator's. This distinction matters because the whole method of this chapter -- writing a single complicated fraction as a sum of simple fractions, each with a small piece of the original denominator underneath it -- only ever applies directly to a proper fraction. An improper fraction is reduced to a proper one first by ordinary polynomial long division, $\dfrac{f(x)}{g(x)}=Q(x)+\dfrac{R(x)}{g(x)}$ with $\deg R<\deg g$, and it is the remainder fraction that is then decomposed (§7.3).

Once a fraction is proper (and in lowest terms, with the denominator fully factored over the reals into linear and irreducible-quadratic pieces), five rules cover every case that can arise:

1. A non-repeated linear factor $(ax+b)$ contributes a term $\dfrac A{ax+b}$.
2. A linear factor repeated $n$ times, $(ax+b)^n$, contributes a full chain $\dfrac{A_1}{ax+b}+\cdots+\dfrac{A_n}{(ax+b)^n}$.
3. A non-repeated irreducible quadratic factor $(ax^2+bx+c)$ contributes a term with a linear numerator, $\dfrac{Ax+B}{ax^2+bx+c}$.
4. A quadratic factor repeated $n$ times contributes the chain $\displaystyle\sum_{i=1}^n\dfrac{A_ix+B_i}{(ax^2+bx+c)^i}$.
5. An improper fraction is divided first; its proper remainder is then handled by rules 1--4.

In every rule, the unknown constants are pinned down the same way: multiply through by the whole denominator to get a polynomial identity, then either substitute a convenient value of $x$ (a root of one of the linear factors makes every other term vanish) or compare the coefficients of matching powers of $x$ on the two sides.