Partial Fractions

Understanding Partial Fractions

  • Remember that Partial Fractions are a way of expressing a given fraction as the sum of simpler fractions.

  • It is necessary to factorise the denominator of the fraction. It is important that this is done before splitting the fraction into parts.

  • The method employed for the division of the fraction into simpler fractions depends on whether the factors in the denominator are linear, repeated linear, or quadratic.

Linear Factors

  • When dealing with linear factors, it can be assumed that the numerator will be a constant for each fraction.

  • Setting up equations by equating numerators and finding common denominators is an essential step.

  • Substitute designed values to get a system of linear equations which can be solved simultaneously to find the constants.

Repeated Linear Factors

  • When dealing with fractions that have repeated linear factors, the denominator for each fraction will be the factor to the power of the number of repetitions.

  • An important point to remember is to set up each fraction with the same factor, but the power of the denominator increases by 1 each time.

Quadratic Factors

  • Given that Quadratic factors cannot be further factorised, they are treated as a single factor in partial fractions.

  • The corresponding fraction should be set up with a linear numerator.

  • When the denominator is a quadratic factor, we need to consider the numerator as (Ax+B) where both A and B are constants to be found.

Integration and Partial Fractions

  • It should be noted that partial fractions are often used in integration problems.

  • When a complicated rational function needs to be integrated, it will be much easier to do so if it has been first split into simpler fractions.

  • Integrating these simpler partial fractions is easier and less error-prone than integrating the original fraction.

Remember, practise is key. Spend a good amount of time solving various problems involving partial fractions to become familiar with them. This will help in easily spotting and efficiently solving questions involving partial fractions in the exam.