7.3: Partial Fraction Decomposition

Definitions

Previously, we have written rational expressions in the form of:

These expressions can also be written as the sum of two or more rational expressions. For example:

The entire right side is known as the partial fraction decomposition, and each fraction is a partial fraction.

The skill of partial fraction decomposition will be used in calculus.

Process of Decomposition

1. Divide if the rational expression is improper. In other words, if the degree of the numerator is greater than or equal to the degree of the denominator, then divide the expression. Complete the remaining steps with the proper rational expression.

2. Factor the denominator.

3. Separate Factors. The following factor must be written as follows:

4. Check your work by combining the partial fractions or graphing the two expressions.

Examples

1.

Multiply by the lowest common denominator.

Since the left side of the equation has no "x" term, (A + B) = 0. (A - B) corresponds to the constant value of 1.

Solving for A and B results in:

 Substitute A and B into the original expression to solve for the partial fraction decomposition:

2.

3.

Since this rational expression is improper, the numerator must be divided by the denominator. This results in:

Now the proper rational expression can be decomposed as normal.

Other Resources

If you would like extra help, the following links provide explanations and examples.

Videos: 

1. Kahn Academy     

2. Patrick    

3. MIT

Text: 

1.Purple Math     

2. Paul 

3. Math World