The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This essential trains for: AMC-10, AMC-12, AIME.

Partial fraction decomposition is a technique used in summation and integration.

The purpose is to break down a term expressed as an algebraic fraction into simpler terms. The simpler terms may then exhibit a pattern of cancellation among successive terms that allows us to telescope the sum. In the case of integration, it helps distribute the integral into two simpler integrals.

Example: Compute the sum

equation

First, notice the pattern that yields the sequence of denominators:

equation

The general term of this sum has the form:

equation

Let us try to write this term as a sum of simpler algebraic fractions:

equation

Perform the addition of algebraic fractions:

equation

Find A and B by identifying the polynomials that result at the numerator:

equation

equation

equation

Therefore:

equation

and the decomposition in partial fractions is:

equation

Now the sum can be written in the form:

equation

The pattern of term cancellations is very obvious. The sum results:

equation

However, the pattern of term cancellation is not always as simple and it is always a good practice to write methodically a few terms from the start and a few terms from the end and use accuracy in determining which terms exactly cancel out.

In a problem solving context, the denominator may be expressed as a polynomial expression that needs to be factored out before attempting the decomposition. As in:

equation

where the degree n of the polynomial at the numerator is lower than the degree m of the polynomial at the denominator.

Since a polynomial can always be factored into linear and at most quadratic factors, we can attempt a decomposition in simple fractions with numerators that are constant or at most linear polynomials.

Example:

equation

We notice that the factors of the denominator cannot be factored into linear terms with real coefficients since the discriminants of the two quadratic expressions are negative.

The numerators of the simple fractions can be at most linear.

Assume the decomposition:

equation

and proceed to identify the polynomials at the numerators:

equation

equation

where from:

equation

equation

equation

equation

with the solution:

equation

Finally, the decomposition is:

equation

And now what?

We still have to make the crucial observation that:

equation

which helps us to finally telescope the sum:

equation

After canceling the terms:

equation