Partial fractions

Understanding Partial Fractions

  • Partial fractions allows for the separation of more complicated rational expressions into simpler fractions that are easier to work with, specifically designed for integration and differentiation.
  • A rational expression or function can be defined as the ratio of two polynomials. The method of partial fractions helps to break these functions into simpler components.
  • In order to apply the method, the degree of the numerator polynomial should be less than the degree of the denominator polynomial. If it’s not, you should use polynomial division first to adjust the expression.

Procedure For Partial Fractions

  • When the denominator factors into linear factors (i.e. terms of x to the power one), each factor will generate a partial fraction of the form A/Bx + C where A and C are constants.
  • When the denominator contains repeated linear factors - say, (x-a)^n - each repeat generates an additional term in the form A/(x-a)^k where k ranges from 1 to n.
  • When the denominator contains irreducible quadratic factors, that is the factor cannot be simplified further, these will lead to terms in the form (Ax+B)/(quadratic factor).
  • The coefficients A, B, … etc for each term are found by equating the original function with the sum of the partial fraction terms, and solving the resulting system of equations.

Integration & Partial Fractions

  • The method of partial fractions can be used to express a rational function as a sum of simpler fractions which can be easily integrated.
  • If the integrand is a rational function, it may be rewritten as a sum of partial fractions, each of which is easily integrable.
  • The integration of most rational functions can thus be reduced to the integration of a polynomial, a constant over a linear function, or a linear function over a quadratic function.

Applications of Partial Fractions

  • Partial fractions are a key part of more advanced calculus, used often in integration techniques and differential equations solving methods.
  • They allow for handling of rational functions in a simpler and more manageable fashion.
  • In physics or engineering, they can simplify calculations related to system dynamics models, or in circuits with capacitors and inductors.