In Calculus when working with Integrals, Laplace Transforms, etc, there’s a process called Partial Fractions Decomposition that helps a lot when we’re doing something and involves rational functions. To begin with it’s important to understand the definition of Proper Fraction and Improper Fraction.

A Proprer Fraction is a fraction whose numerator is smaller that the denominator, here’s the definition:

While a Improper Fraction is exactly the opposite, a fraction whose numerator is bigger than the denominator, and this kind of fractions can be a problem when working with the Partial Fraction Decomposition method, sometimes we’il need to to a simple polinomial division in order to write it as an proprer fraction subject to a Partial Fraction Decomposition method. Here we have the definition a Improper Fraction:

There’s a fundamental moto that is important to get: “Any Rational Proper Function can be decomposed in a sum of simple fractions”. For instance:

An example of a rational function that isn’t proper is this one we’il work later on:

So in the most common case when we can decompose the denominator in simple factors (factors without multiplicity bigger than 1) we generalize the Partial Fraction Decomposition method this way:

When we have a denominator decomposed into factors with some kind of multiplicity we simply repeat the factors:

**Proper Fractions**

The better way of understanding it is by practising lets do it:

Notice that i factored the denominator by using the Sum/Product method, but you can do it as you want, for instance, using Bhaskara’s Formula. Now we just have to find the least common multiple between the two partial fractions and simplify them:

Now we have to work with this iguality:

In order to find values of A and B, we have to let ‘x’ be the roots of the expressions multiplying A and B, once at time. To find B, let x=-2:

Now in order to find A, we do the same process, let x=1:

So finally we have:

**Improper Fractions**

As said before when having an improper fraction, a fraction whose numerator is greater than the denominator we have to proceed to a polinomial division, to understand how we can write a fraction using the division quocient, the rest and the divisor: