Partial Faction Decomposition (Included in 7.3)

Partial fraction decomposition is essential to success in Calculus next year.  PFD involves breaking down a rational faction with a polynomial denominator into the sums of smaller fractions.

Before "decomposing" a fraction, you first have to take the degrees of the numerator and denominator of the rational function into consideration.

• If the degree of the numerator is greater than that of the denominator (improper fraction), divide the numerator into the denominator by long division.  This will result in a polynomial followed by the remainder of the long division divided by the original denominator.  Then proceed to decompose the latter using the steps that will be explained later.
• If the degree of the numerator and the denominator equal, follow the steps above and use long division to break the function down further.
• If the degree of the numerator is less than that of the denominator (proper fraction), proceed using the steps below.

Here is an example:

   First, factor the denominator. 

Next, make this fraction equal to a sum of two fractions with the factored denominators.

Working with the right side, proceed as you would to find a common denominator to add the fractions together.  Add the like terms and begin to simplify the equation.














As you can see the  term has a coefficient of 1 and therefore  equals 1.  Consequently,  must then equal 5.  Create a system of equations from here and solve.







Your final answer:




Side note- In cases where the denominator has repeated factors such as , be sure to decompose as followed:





Thanks for reading!

Julia Wilkins