Partial Fraction Decomposition

Partial Fraction Decomposition

We can easily combine two rational expressions

into a single rational expression

However, sometimes it is useful to be able to take a large rational expression like

and to figure out how to break it down into the smaller rational expressions

The method is called partial fraction decomposition.

We consider rational expressions of the form

where

If this is not the case, just do long division first to obtain a polynomial plus a rational expression of the above form.

I will demonstrate the method using some examples.

Case 1: Linear factors

Let .

Then the expression decomposes into

Notice that we need a fraction for each linear factor and its powers. In general

Example: Decompose

We first write

We need only look at the numerators

There are two ways to get those constants

, and .

We could multiply out the left side and equate coefficients of the powers of

. Then we get

Let

. Then we get

Let

. We get

We could have substituted in any values of

we wanted but you can see there are better and worse choices.

Case 2: Irreducible quadratic factors.

When you wish to decompose q rational expression with an irreducible quadratic factor, i.e. a quadratic factor that does not factor into linear factors of real numbers, then you need to handle it differently.

Here is how the decomposition would look.

Notice we need a linear factor, not a constant, over the quadratic denominator.

Example: decompose

We first write

Then consider the numerator

Let

. We get

. We get

Let

. We get

So

Green Question: (CEMC vol.8) Given that

is an integer, for how many values of will

also be an integer?

Red Question: Find the sum: