Partial Fractions

The central idea

A proper rational function can be decomposed into simpler fractions after its denominator is factored over the chosen field. Distinct linear factors use $A/(x-a)$; repeated factors require every power; irreducible quadratic factors use linear numerators. An improper rational function must first undergo polynomial division.

Definitions, hypotheses, and notation

Coefficient values can be solved by evaluating at roots only for the portions those roots isolate; repeated factors and irreducible quadratics often require derivative or coefficient comparison steps. Factoring over the reals keeps quadratics such as $x^2+1$ intact, while a complex decomposition would look different.

Domain intervals matter after integration. Logarithmic constants can differ across disconnected intervals separated by denominator zeros. Absolute values preserve the derivative $1/(x-a)$ on either side, but the original rational function still remains undefined at the pole.

Conceptual meaning

The decomposition rewrites one complicated rational function as a sum whose antiderivatives are logarithmic, reciprocal-power, or inverse-trigonometric forms. It is an algebraic identity on the common domain, not an approximation.

A dependable method and decision rule

1. Compare degrees and perform long division if necessary.
2. Factor the denominator completely.
3. Write the full decomposition template, including repeated powers.
4. Clear denominators and solve coefficients by strategic substitution or coefficient comparison.
5. Integrate each term and differentiate to verify.

Fully worked example

Common mistakes and why they fail