Partial Fraction Decomposition

Partial fraction decomposition breaks a complicated rational function into a sum of simpler ones: $$\frac{P(x)}{Q(x)} = \frac{A_1}{(x - r_1)} + \frac{A_2}{(x - r_2)} + \cdots$$ Intuition: Just as you can factor an integer into primes, you can factor a rational function into "atomic" fractions. Each factor in the denominator contributes one simple fraction. This makes integration, summation, and generating function extraction straightforward. Algorithm for distinct linear factors: 1. Factor the denominator: $Q(x) = (x-r_1)(x-r_2)\cdots(x-r_n)$ 2. Write: $\frac{P(x)}{Q(x)} = \frac{A_1}{x-r_1} + \frac{A_2}{x-r_2} + \cdots$ 3. Multiply both sides by $Q(x)$ and solve for $A_i$ (plug in $x = r_i$) Concrete example: Decompose $\frac{1}{k(k+2)}$. $$\frac{1}{k(k+2)} = \frac{A}{k} + \frac{B}{k+2}$$ Multiply by $k(k+2)$: $1 = A(k+2) + Bk$. Set $k=0$: $1 = 2A$, so $A =$$\frac{1}{2}$. Set $k=-2$: $1 = -2B$, so $B = -$$\frac{1}{2}$. Result: $\frac{1}{k(k+2)} = \frac{1}{2}\left(\frac{1}{k} - \frac{1}{k+2}\right)$. When to use: Simplifying rational expressions for telescoping sums, integration, inverse Laplace transforms, and generating function extraction. A standard step in solving recurrences via generating functions. This is a fundamental technique — essential plumbing for analysis, combinatorics, and probability alike.