Partial fraction decomposition calculator with steps

On many occasions when we work with algebra problems, we have to deal with relatively complex rational functions. The Partial Fractions Calculator with steps that we present here will allow you to decompose a rational function into simple fractions with just three simple steps:

1. Enter the expression of the numerator.
2. Enter the polynomial of the denominator.
3. Press the green “Calculate” button. The solution explained step by step will be displayed automatically.

Rational functions are basically polynomial or algebraic fractions, in which we can find polynomials either in the numerator or in the denominator. To simplify a rational function there is a technique called Partial Fractions, which consists of decomposing a rational function into the polynomial sum of simple fractions.

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How to do partial fraction decomposition

In order to simplify a rational function using the method of partial fractions or decomposition into simple fractions, we must apply the following steps:

Example:

1. Inspect the expression to see if it is a proper or improper fraction. If the expression is an improper fraction, that is, the degree of the denominator is less than the degree of the numerator, then you must perform polynomial long division. For our problem, the degree of the denominator is 3, and the degree of the numerator is 0, so we’re good to go. In the case of our example it is a proper fraction.
2. Factor the numerator and denominator of the expression (or the remainder, if one was obtained in step 1), canceling out any factors that the numerator and denominator have in common. In the case of our example, it is no longer possible to factorize more than it is.
3. From the factors in the denominator, write the appropriate partial fractions with unknown coefficients in the numerator. The form of these terms is found in your textbook and many other online resources, so I won’t explain them again here. Suffice it to say that our partial fraction expansion will look like this:

4. Multiply both sides of the equation by the denominator of the original expression and simplify. In the case of the example that we are developing, this will be as follows:

6. Solve the resulting equation system to obtain the values of the coefficients, and then substitute them in the expression obtained in step 3, thus achieving the decomposition into partial fractions. For the example that we are dealing with, the result of the decomposition into simple fractions is as follows: