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**No Repeated Roots**

Let's say we have a proper function

The first step is to factor the denominator

where

where

We can easily generalize this to a solution for any one of the unknown coefficients:

This method is called the "cover-up" method because
multiplying both sides by

#### Example 1

In this example, we'll work through the partial fraction expansion of the ratio of polynomials introduced above. Before doing a partial fraction expansion, you must make sure that the ratio you are expanding is proper. If it is not, you should do long division to turn it into the sum of a proper fraction and a polynomial. Once this is done, the first step is to factor the denominator of the function:

Now, we set this factored function equal to a sum of smaller fractions, each of which has one of the factored terms for a denominator.

To find the alpha terms, we just cover up the
corresponding denominator terms in

We now have our completed partial fraction expansion:

###
**Repeated Roots**

When the function

Somewhat more special care must be taken to find the partial fraction expansion. The non-repeated terms are expanded as before, but for the repeated root, an extra fraction is added for each instance of the repeated root:

All of the alpha constants can be found using the non-repeated roots method above. Finding the beta coefficients (which are due to the repeated root) has the same Heaviside feel to it, except that this time we will add a twist by using the derivative to eliminate some unwanted terms.

Starting off directly with the cover-up method, we can find

Now that we have "covered up" the

To find the other values of the beta coefficients, we can
take advantage of the derivative. By taking the derivative
of Equation 14 (with respect to

Generalizing over this pattern, we can continue to take derivatives to find the other beta terms. The solution for all beta terms is