Partial Fractions

Partial Fractions

Understanding Partial Fractions

  • Partial fractions involve the decomposition of a complex fraction into simpler fractions. This technique is useful in calculus and differential equations.
  • The numerator must be of a lower order (degree) than the denominator for a fraction to be split into partial fractions.
  • It is necessary to perform long division before partial fractions can be used if this condition is not met.

Types of Partial Fractions

  • Proper Fractions: The degree of the numerator is less than the degree of the denominator.
  • Improper Fractions: The degree of the numerator is greater than or equal to the degree of the denominator.
  • Linear Partial Fractions: The factors of the denominator are linear (i.e., they are to the power of 1).
  • Repeated Linear Partial Fractions: The factors of the denominator are the same linear factor repeated.
  • Irreducible Quadratic Factors: The denominator can have factors that are irreducible quadratics.

Techniques to Solve Partial Fractions

  • Factorise the denominator: The first step in splitting fractions into partial fractions is to factorise the denominator.
  • Split into partial fractions: After factorising the denominator, express the fraction as the sum of simpler fractions with unknown numerators.
  • Finding the unknowns: Equate coefficients or substitute suitable values to find the values of the unknowns.

Benefits of using Partial Fractions

  • Simplify integration and differentiation: Calculating the integral or derivative of multiple simple fractions can be easier than calculating the integral or derivative of a single complex fraction.
  • Simplify algebraic manipulation: Partial fractions can make solving equations or rearranging expressions easier.

Common mistakes

  • Not fully factorising the denominator: Ensure that the denominator is fully factorised into linear or irreducible quadratic factors to ensure the correct form of the partial fractions.
  • Overlooking repeated roots / factors: If a factor of the denominator is repeated, a partial fraction is needed for each occurrence of the factor.
  • Incorrect degree of the numerator: Make sure the degree of the numerator in each partial fraction is one less than the degree of the factor in the denominator.