Early in Algebra you learn how to combine "simple'' fractions into a "more complicated'' one. Here is a typical example:
The Method of Partial Fractions does the opposite: It dissects a complicated fraction into a sum of simple fractions. While this is a little more complicated than going the other direction, it is also more useful. Major applications of the method of partial fractions include:
A simple fraction is a fraction with a simple denominator. The first step consists of detecting the factors (the building blocks) of the given denominator. The Fundamental Theorem of Algebra tells us what is possible: Every polynomial can be factored into linear factors (degree 1 polynomials) and irreducible polynomials of degree 2.
There are different methods to decide:
Since the discriminant (the expression under the radical) is negative, the polynomial is irreducible!