Partial Fractions

• Every proper rational function can be expressed as a sum $\frac{P(x)}{Q(x)}=F_1(x)+F_2(x)+\cdots +F_n(x)$ where $F_1(x),F_2(x),\dots ,F_n(x)$ are rational functions of the form $\frac{A}{(ax+b{)}^k}$ or $\frac{Ax+B}{(a{x}^2+bx+c{)}^k}$ in which the denominators are factors of $Q(x)$.
  • The sum is called the partial fraction decomposition of $P(x)/Q(x)$, and the terms are called partial fractions

Distinct Linear Factors

Suppose that we can factor $Q(x)$ as $Q(x)=(a_{1}x+b_1)(a_{2}x+b_2)\cdots (a_{n}x+b_n)$ with no factor repeated. In this case the partial fraction decomposition of $P(x)/Q(x)$ takes the form $\frac{P(x)}{Q(x)}=\frac{A_1}{(a_{1}x+b_1)}+\frac{A_2}{(a_{2}x+b_2)}+\cdots +\frac{A_n}{(a_{n}x+b_n)}$

Repeated Linear Factors

Suppose the complete factorization of $Q(x)$ contains the linear factor $ax+b$ repeated $k$ times; that is, $(ax+b{)}^k$ is a factor of $Q(x)$. Then, corresponding to each such factor, the partial fraction decomposition for $P(x)/Q(x)$ contains $\frac{A_1}{ax+b}+\frac{A_2}{(ax+b{)}^2}+\cdots +\frac{A_k}{(ax+b{)}^k}$