Chapter 7.4, Problem 69E

To determine

To calculate:

Describe two ways of solving for the constants in a partial fraction decomposition.

Answer to Problem 69E

The two methods are described for solving the constants in a partial fraction decomposition.

Explanation of Solution

Given information:

We need to give an explanation of the ways of solving for the constants in a partial fraction decomposition.

First method:

Partial fraction decomposition is the process of starting with the simplified answer and taking it back apart of decomposing the final expression into its initial polynomial fractions.

To decompose a fraction, you first factor the denominator. Then you write one of the factors for each of the denominators. As we do not know what the numerators are yet, so you assign variables for these unknown values.

For example, $\frac{A}{x}+\frac{B}{x+1}$

Then you set this sum equal to the simplified result. Then multiply through the common denominator to get rid of all denominators.

Multiply things out and group the x-terms and the constant terms. For the two sides to be equal, the coefficients of the two polynomials must be equal.This creates a system of equations that we can solve easily and find the original fractions.

Second method:

There is another method for solving the equations. Since the equation is supposed to be true for any value of x, we can pick useful values of x, trial and error, and find the values for the variables.