Partial fraction decomposition

Partial fraction decomposition expresses a rational function as a sum of simpler fractions whose denominators are powers of irreducible polynomial factors.

Rational function: A quotient of two polynomials, with a nonzero denominator. Partial fraction decomposition applies to rational functions and rewrites them as sums.
Integration of rational functions: The process of finding antiderivatives of quotients of polynomials. Partial fractions reduce many such integrals to standard logarithmic and power-rule forms.
Polynomial: An expression formed from variables and coefficients using addition, subtraction, and nonnegative integer powers. Numerators and denominators in a rational function are polynomials.
Heaviside cover-up method: A shortcut for finding coefficients in certain partial fraction decompositions with distinct linear factors. It can replace full coefficient matching in simple cases but does not handle every factor pattern.
Polynomial factorization: The expression of a polynomial as a product of lower-degree polynomials. Factoring the denominator identifies the terms needed in the decomposition.
Improper integral: An integral with an unbounded interval or an integrand that is unbounded within the interval. Decomposed terms can make convergence near poles or at infinity easier to analyze.
Degree of a polynomial: The highest exponent of the variable with a nonzero coefficient in a polynomial. Degree determines whether division is needed before the denominator is decomposed.
Continued fraction: An expression formed by nesting integer or rational terms in repeated denominators. Continued fractions provide a different representation of rational expressions, not a sum over denominator factors.
Irreducible polynomial: A nonconstant polynomial that cannot be factored into lower-degree polynomials over a specified coefficient field. Each irreducible denominator factor determines a distinct fraction form.
Laplace transform: An integral transform that maps a function of time to a function of a complex variable. Inverse transforms of rational expressions are often found by splitting them into simpler terms.

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