Problem 49 - College Algebra

Question

Explain how to find the partial fraction decomposition of a rational expression with a prime quadratic factor in the denominator.

Step-by-Step Solution

Answer

The partial fraction decomposition of a rational expression with a prime quadratic factor in the denominator demands factoring the denominator, setting up partial fractions (\(\frac {Ax + B}{ax^2 + bx + c}\)), equating coefficients, and solving for the unknowns \(A\) and \(B\).

Step 1: Factoring the Denominator

The first thing to do is to factor the denominator of the given rational expression. By doing this, it will be easier to set up the partial fractions. If the quadratic factor is prime, it cannot be factored, and should be left as is for the moment.

Step 2: Set Up the Partial Fractions

The next step involves setting up the partial fractions. Each fraction will contain one of the factors from the denominator of the original fraction. If the denominator contains a prime quadratic factor of the form \(ax^2 + bx + c\), then the corresponding partial fraction should take the form \(\frac {Ax + B}{ax^2 + bx + c}\), where \(A\), \(B\) are constants to be determined.

Step 3: Equating Coefficients

The next stage is to equate coefficients from the original fraction to those from the decomposed fraction. Before you can do this, you need to combine the partial fractions on the right side and rewrite as a single fraction. Then, by comparing coefficients, we get an equation that can be solved for the unknown constants \(A\) and \(B\).

Step 4: Solve for Constants

The final step involves solving the equation(s) obtained from equating coefficients. These solutions give the values of constants \(A\) and \(B\), which complete the decomposition of the original fraction.