What we desire to do in this section is to begin with rational expressions & ask what simpler rational expressions did we add and/or subtract to obtain the original expression. The procedure of doing it is called partial fractions & the result is frequently called the partial fraction decomposition.
The procedure can be a little long and on occasion messy; however it is really fairly simple. We will begin by trying to find out the partial fraction decomposition of,
P ( x )/ Q ( x )
Where both P(x) & Q(x) are polynomials & the degree of P(x) is smaller than the degree of Q(x). Partial fractions can just be done if the degree of the numerator is firmly less than the degree of the denominator. i.e. important to remember.
Hence, once we've determined which partial fractions can be performed we factor the denominator as wholly as possible. Then for each of the factor in the denominator we can utilize the following table to find out the term(s) we pick up in the partial fraction decomposition.
Notice that the first & third cases are actually special cases of the second & fourth cases respectively if we consider k = 1 . Also, it will entirely be possible to factor any polynomial down in product of linear factors ( ax+ b ) and quadratic factors ( ax^{2} + bx+ c ) some of which might be raised to a power.