Partial fractions decomposition is the opposite of adding fractions, we are trying to break a rational expression into simpler fractions. It takes a lot of work, but is extremely useful with integrals for instance (simplification can be a good strategy). We start by factoring the denominator (if the numerator order is higher than the denominator we start with long division), then we write the partial fraction for each of the factors (watch out for high order factors), multiply and solve for the coefficients using the factors zeros. Step by step examples can be really helpful here.

Let’s start with an example (

click here):

Here’s a more advanced example with high order factors (

click here):

From here simply solve the equation and plug in the solutions to get the partial fractions.

Here’s an example where the order of the numerator is higher than the denominator (

click here):

Cheers,

Michal