Partial Fractions | Quadratic and Repeated Linear Factors

So far we have performed partial fraction decomposition with expressions that have had linear factors in the denominator, and we applied numerators A, B, or C representing constants. Now we will look at an example where one of the factors in the denominator is a quadratic expression that does not factor. This is referred to as an irreducible quadratic factor. In cases like this, we use a linear numerator such as Ax+B, Bx+C etc. Then partial Fraction Decomposition Form for Repeated Factors: A factor is repeated if it has multiplicity greater than 1. If the repeated factor is linear, then each of these rational expressions will have a constant numerator coefficient. If the factor is to the power of 3. Therefore, 3 rational expressions are needed in the partial fraction decomposition, with each of the denominator is raised to a different positive integer power up to 3, wih 1, 2 and 3 respectively. Since it's linear factor, each of these rational expressions will have a constant numerator coefficient.