Partial Fraction Decomposition

Solved Problems

Solution.

The partial fraction decomposition has the form

We combine the partial fractions on the right side into a single fraction and then equate the numerators on both sides. This yields:

Hence we obtain the following system of equations:

Solving it, we find the unknown coefficients:

The partial fraction decomposition is given by

Solution.

First we factor the cubic function in the denominator:

As the denominator contains only linear factors, the decomposition has the form

Combining the partial fractions on the right side into a single fraction and equating the numerators on both sides, we have:

The system of equations for the unknown coefficients is written as

It has the following solution:

so the decomposition of the original rational function is expressed by the equation:

Solution.

Given the irreducible quadratic factor $$ in the denominator, we write the partial fraction expansion in the form

The equation for the unknown coefficients is given by

or

We obtain the system of equations:

This yields:

Hence, the partial fraction decomposition is written as

Solution.

Note that the denominator $$ contains the irreducible quadratic factor $$ and the linear factor $$ with multiplicity $$ Hence, the partial fraction decomposition is written as

To determine the unknown coefficients $$ we clear fractions multiplying both sides by the denominator. This gives

By equating coefficients of similar powers on both sides, we have

We solve the resulting system and find

Hence, the partial fraction decomposition of the rational function is given by

Solution.

The denominator has the irreducible quadratic factor $$ Hence, the partial fraction decomposition has the form

By equating the numerators on both sides of the equation, we have

We get the following system of equations:

It has the solution

Hence, the partial fraction decomposition is given by