Problem 48 - 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

To find the partial fraction decomposition of a rational expression with a prime quadratic factor in the denominator, the steps are factoring the denominator, setting up the equation with partial fractions, equating coefficients to form a system of equations, and finally solving this system to find the values of the constants. These constants are then substituted back into the partial fractions to get the final solution.

Step 1: Factorize the Denominator

Initially, the denominator of the given rational expression should be factorized. If the expression in the denominator can be factorized to have irreducible quadratic and/or linear terms then it is possible to proceed to the next step.

Step 2: Set Up Equation with Partial Fractions

Following factoring, it is crucial to set up the equation correct. For every linear term in the factorization, terms of the form A/x should be added where A is the constant coefficient. Meanwhile, for each irreducible quadratic term, expressions of the form (Bx+C)/x² should be included, where B and C are constants.

Step 3: Equate Coefficients

After setting up the equation, multiply both sides by the factored denominator to clear the fraction. Upon resulting in a polynomial equation, equate the coefficients of corresponding powers on both sides of the equation to determine values for A, B and C. This will result in a system of linear equations in the constants.

Step 4: Solve the System of Equations

The last crucial step is to solve for the constants A, B and C. After determining their values, substitute them back into the partial fraction decomposition from step 2.