Partial fractions

When a rational function’s denominator can be factored into linear pieces, we can rewrite the whole fraction as a sum of simpler fractions whose denominators are those pieces. This is called partial‑fraction decomposition. The key idea is to assume the original fraction equals a sum of unknown constants over each factor (or over powers of a repeated factor), then clear denominators and match coefficients of like powers of \(x\). Solving the resulting linear equations gives the constants, completing the decomposition. This technique turns a complicated rational expression into a combination of easy‑to‑integrate or simplify pieces.