Problem 55 - College Algebra Essentials

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 partial fraction decomposition of a rational expression with a repeated linear factor in the denominator involves breaking the complex fraction into fractions with the repeating factor and its square. The procedure involves setting up an equation and solving for the constants in the decomposed fraction, which are then substituted into the general formula for the decomposed fraction.

Step 1: Identify the rational expression

The first thing to do is write down the rational expression that needs to be decomposed. For example, let the rational expression be \(\frac{3x^2 - 2x + 1}{(x-1)^2}\).

Step 2: Understand partial fraction decomposition

Partial fraction decomposition is a process of breaking down a complex fraction into simpler fractions. In the case of our expression, with a repeated linear factor in the denominator \(x - 1\), the decomposed form will be the sum of fractions in the form \(\frac{A}{x - 1}\) and \(\frac{B}{(x - 1)^2}\), where A and B are constants to be determined.

Step 3: Set up the equation

To find the values of A and B, set the original rational expression equal to the sum of the simpler fractions. This should give you an equation: \(\frac{3x^2 - 2x + 1}{(x - 1)^2} = \frac{A}{x - 1} + \frac{B}{(x - 1)^2}\). You should then multiply both sides by the common denominator to clear the fractions which gives \(3x^2 - 2x + 1 = A(x - 1) + B\).

Step 4: Solve for constants

To solve for A and B, choose convenient values for x. Starting by setting \(x=1\) to remove A from the equation and find B. Then choose another value for \(x\) and substitute the values of B found earlier to find A.

Step 5: Formulate the final decomposed fraction

Substitute the calculated values of A and B back into the partial fraction decomposition \(\frac{A}{x - 1} + \frac{B}{(x - 1)^2}\). This gives us a final decomposed form of the original rational expression.