Partial Fraction Decomposition is predicated on the premise that a rational function — a quotient of two polynomials where the numerator is of lesser degree than the denominator — can be expressed as a sum of simpler fractional components. This theorem is instrumental in the realm of integral calculus for its facility in decomposing complex integrands into more tractable forms.
The proof of the theorem is an exercise in algebraic manipulation and relies on several fundamental tenets of algebra:
Polynomial Long Division: Should the rational function not be proper, i.e., the degree of the numerator is greater or equal to that of the denominator, polynomial long division is employed to render it proper.
Factorization of the Denominator: The denominator is decomposed into its irreducible polynomial factors, which are either of first degree (linear) or second degree (quadratic) that do not further factorize over the field of real numbers.Post too long. Click here to view the full text.