Problem 47 - College Algebra

Question

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

Step-by-Step Solution

Answer

The main steps to finding the partial fraction decomposition of a rational expression with a repeated linear factor in the denominator are: Define the general form of the decomposition, solve for the coefficients, and express the final decomposition. The complexity will depend on the degree of the repeated factor and the nature of the polynomial in the numerator.

Step 1: Identify The Form of The Rational Expression

First, observe the form of the rational function given. If the denominator has a repeated linear factor, then it should appear as \( \frac{P(x)}{(ax+b)^n} \), where \( P(x) \) is a polynomial of degree less than \( n \), \( ax+b \) is the repeated linear factor, and \( n \) is the number of times the factor is repeated.

Step 2: Formulate the General Form of Decomposition

Once the form has been identified, formulate the general form of the decomposition. This general form should include several fractions in addition, each with a constant in the numerator and the factor in the denominator raised to a power. The powers on the factor in the denominators should descend from \( n \) down to 1. It should look like: \( \frac{A}{ax+b} + \frac{B}{(ax+b)^2} + \frac{C}{(ax+b)^3} + ... + \frac{Z}{(ax+b)^n} \) where \( A, B, C...Z \) are constants to be determined.

Step 3: Solve for Coefficients

Next, equate the original rational expression with the general form derived. Cross multiply to get rid of the denominators and isolate terms to form a system of equations. These equations can then be solved to find the values of \( A, B, C...Z \).

Step 4: Express Final Decomposition

After obtaining the values of \( A, B, C...Z \), substitute these values back into the general form formulated in Step 2. This obtains the partial fraction decomposition of the original rational function. Simplify if necessary.