Problem 57 - College Algebra

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

The method of partial fractions involves expressing the given fraction as sums of simpler fractions. In the case of a rational expression with a repeated, prime quadratic factor in the denominator, the decomposed form should look like \( \frac{A}{x-a} + \frac{Bx+C}{(x-a)^2+b} \), with A, B, C calculated from the system of equations constructed by equating coefficients for the powers of x after clearing the fraction in the initial equation.

Step 1: Identifying Proper Format

The first step is to make sure the rational function is in the proper form. If the degree of the numerator is greater than or equal to the degree of the denominator, divide the numerator by the denominator to get a polynomial plus a proper rational function.

Step 2: Writing Out the Decomposition Form

Express the rational function as a sum of fractions, with each denominator in the sum being the distinct linear factors and repeated linear factors of the denominator of the given rational expression. In other words, set up the equation: \( \frac{f(x)}{(x-a)^2 + b} = \frac{A}{x-a} + \frac{Bx+C}{(x-a)^2 + b}\). A, B, C are coefficients to be determined.

Step 3: Clearing the Fraction

Clear the fraction by multiplying the entire equation by the common denominator. This removes the fraction.

Step 4: Equating Coefficients

Arrange the remaining equation in terms of powers of x. Then equate the coefficients for the powers of x on both sides of the equation and solve the resultant system of equations to find the unknown coefficients A, B, C.

Step 5: Writing the Solution

Once you have the coefficients, substitute them back into the equation obtained in step 2. This gets you the partial fraction decomposition of the original rational function.