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Business Mathematics and Statistics · Ch 2 — Algebra (Partial Fractions, Permutations, Combinations, Mathematical Induction, Binomial Theorem)

Partial Fractions — Resolving into Partial Fractions

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Partial Fractions — Resolving into Partial Fractions

Once a rational fraction is proper, it can be broken into a sum of simpler fractions whose denominators are the individual factors of the original denominator. The exact form of each simple fraction depends on the type of factor.

Case 1 — Non-repeated linear factors. If g(x)=(x−a)(x−b)g(x) = (x-a)(x-b) with a≠ba \neq b, write

f(x)(x−a)(x−b)=Ax−a+Bx−b\frac{f(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}

Case 2 — Repeated linear factor. If g(x)g(x) contains a factor (x−a)2(x-a)^2, that factor contributes two terms, not one:

f(x)(x−a)2(x−b)=Ax−a+B(x−a)2+Cx−b\frac{f(x)}{(x-a)^2(x-b)} = \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b}

A factor repeated nn times contributes nn separate terms, with denominators (x−a),(x−a)2,…,(x−a)n(x-a), (x-a)^2, \ldots, (x-a)^n.

Case 3 — Irreducible quadratic factor. If g(x)g(x) contains a quadratic factor that cannot be factorised into real linear factors, that factor contributes a term with a linear numerator:

f(x)(x−a)(px2+qx+r)=Ax−a+Bx+Cpx2+qx+r\frac{f(x)}{(x-a)(px^2+qx+r)} = \frac{A}{x-a} + \frac{Bx+C}{px^2+qx+r}

Method. In every case, multiply both sides by the full denominator to clear fractions, then find the unknown constants either by:

  1. Substitution — choosing convenient values of xx (usually the roots of the linear factors) that make all but one unknown vanish, or
  2. Comparing coefficients — expanding both sides and equating the coefficients of each power of xx. …
Definition 1Repeated linear factor

A linear factor (x-a) that occurs more than once in the denominator; each power from 1 up to its multiplicity contributes its own …

Definition 2Irreducible quadratic factor

A quadratic factor with no real linear factors (negative discriminant); it contributes a term with a linear numerator (Bx+C) …

Definition 3Comparing coefficients

A method of finding the unknown constants by expanding both sides after clearing denominators and matching the coefficie …

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