Problem 57 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

To find the partial fraction decomposition of a rational expression with a repeated prime quadratic factor in denominators, set up the expression in the form \( \frac{P(x)}{Q(x)^n} = \frac{A_1}{Q(x)} + \frac{A_2}{Q(x)^2} + ... + \frac{A_n}{Q(x)^n} \), then solve for the constants using suitable values of x from the roots of the quadratic, then substitute the values obtained back into the right side of the equation. This will give the partial fraction decomposition of the rational expression.

Step 1: Setting Up the Equation

Given a rational expression in the form \( \frac{P(x)}{Q(x)^n} \), where \( Q(x) \) is a prime quadratic and \( n \) is the degree of repetition, the equation is written as \( \frac{P(x)}{Q(x)^n} = \frac{A_1}{Q(x)} + \frac{A_2}{Q(x)^2} + ... + \frac{A_n}{Q(x)^n} \)

Step 2: Clarity the Equation

Next is to clear out the denominator by multiplying both sides of the equation by \( Q(x)^n \). This results in \( P(x) = A_1 Q(x)^{n-1} + A_2 Q(x)^{n-2} + ... + A_n \)

Step 3: Solve for Variables

Select suitable values of \( x \) that will simplify the equation and solve for the unknowns \( A_1, A_2, ..., A_n \). The suitable values are usually the roots of the quadratic expression \( Q(x) \).

Step 4:Substitute Values

Substitute the values obtained in the previous step into the right side of the equation from Step 1. This will give the partial fraction decomposition of the rational expression.