Partial Fractions

A rational expression $\dfrac{f(x)}{g(x)}$ is a proper fraction if $\deg f<\deg g$; every proper fraction whose denominator factors into linear and irreducible-quadratic pieces can be written uniquely as a sum of simpler pieces -- its partial fraction decomposition:

1. for each linear factor $(x-a)$ appearing to power $k$ in $g(x)$: $\dfrac{A_1}{x-a}+\dfrac{A_2}{(x-a)^2}+\cdots+\dfrac{A_k}{(x-a)^k}$;
2. for each irreducible quadratic factor $(x^2+ax+b)$ (no real zeros) appearing to power $k$: $\dfrac{B_1x+C_1}{x^2+ax+b}+\dfrac{B_2x+C_2}{(x^2+ax+b)^2}+\cdots+\dfrac{B_kx+C_k}{(x^2+ax+b)^k}$. Finding the constants. After multiplying through by $g(x)$, the unknown constants can be found either by substituting the roots of each linear factor directly (the cover-up shortcut -- e.g. for $\dfrac x{(x+3)(x-4)}=\dfrac A{x+3}+\dfrac B{x-4}$, setting $x=4$ isolates $B$ and $x=-3$ isolates $A$ instantly), or -- needed whenever an irreducible quadratic factor or a repeated factor is present -- by expanding and comparing coefficients of matching powers of $x$ on both sides, often combined with a convenient extra substitution such as $x=0$. …