Decomposing a Rational Function into a Sum of Partial Fractions

Decomposing a Rational Function into a Sum of Partial Fractions

Understanding Rational Functions

  • A rational function is a function that can be written as the ratio of two polynomials where the denominator is not zero.
  • Rational functions are equivalent to rational fractions in algebra, and the processes for simplifying them are similar.

Decomposition into Partial Fractions

  • Decomposing a rational function into a sum of partial fractions involves breaking down a complex fraction into simpler frictions that, when added together, equal the original function.
  • This helps simplify calculations, especially when integrating or differentiating in calculus.

The Decomposition Process

  • Determine the degree of both the numerator and the denominator. If the degree of the numerator is equal to or greater than the denominator, use polynomial division to rewrite the function.
  • Factorise the denominator of the rational function. Each factor corresponds to a partial fraction in the decomposition.
  • Write the expression for the decomposition. Each factor of the original denominator becomes the denominator of a partial fraction.
  • Determine the numerators of the partial fractions by equating the original function with the sum of the partial fractions and solving for the unknown coefficients.
  • Simplify and check your decomposition by adding the partial fractions together. The sum should be equal to the original rational function.

Case Scenarios in Decomposition

  • If a factor in the denominator is a simple linear term (e.g. x+3), its corresponding partial fraction has a constant numerator.
  • If a factor in the denominator is a repeated linear term (e.g. (x+3)^2), the successive partial fractions have numerators that increase in degree (e.g. A/(x+3) + B/(x+3)^2).
  • If a factor in the denominator is a quadratic term (e.g. x^2+1), its corresponding partial fraction has a linear numerator (e.g. (Ax+B)/(x^2+1)).

Example

  • Decompose 2x^2/(x-1)(x^2+1): The denominator has been factored into (x-1)(x^2+1). So, we can write the decomposition as A/(x-1) + (Bx+C)/(x^2+1). Use the method above to solve for A, B, and C.

Note: Having a good grasp of algebraic manipulation is vital for doing well in the decomposition of a rational function into a sum of partial fractions. It requires a solid understanding of polynomial equations, factoring techniques, and solving linear equations.