In Exercises 9–42, write the partial fraction decomposition of each rational expression. x²/(x − 1)² (x + 1)

Verified step by step guidance

1. Identify the denominator and its factors: The denominator is \( (x - 1)^2 (x + 1) \), which consists of a repeated linear factor \( (x - 1)^2 \) and a distinct linear factor \( (x + 1) \).
2. Set up the form of the partial fraction decomposition: For the repeated linear factor \( (x - 1)^2 \), include terms with denominators \( (x - 1) \) and \( (x - 1)^2 \). For the linear factor \( (x + 1) \), include a term with denominator \( (x + 1) \). So, write the decomposition as \( \frac{A}{x - 1} + \frac{B}{(x - 1)^2} + \frac{C}{x + 1} \), where \( A \), \( B \), and \( C \) are constants to be determined.
3. Write the equation equating the original rational expression to the sum of partial fractions: \[ \frac{x^2}{(x - 1)^2 (x + 1)} = \frac{A}{x - 1} + \frac{B}{(x - 1)^2} + \frac{C}{x + 1} \].
4. Multiply both sides of the equation by the common denominator \( (x - 1)^2 (x + 1) \) to clear the denominators: \[ x^2 = A(x - 1)(x + 1) + B(x + 1) + C(x - 1)^2 \].
5. Expand the right-hand side and collect like terms in powers of \( x \) to prepare for solving for \( A \), \( B \), and \( C \) by equating coefficients or substituting convenient values of \( x \).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions with denominators that are factors of the original denominator. This technique simplifies integration and other algebraic operations by breaking down complex fractions into manageable parts.

Factorization of Denominators

Understanding how to factor the denominator into linear and repeated factors is essential. In this problem, the denominator is (x − 1)²(x + 1), which includes a repeated linear factor (x − 1)² and a distinct linear factor (x + 1). Recognizing these factors guides the form of the partial fractions.

Setting Up and Solving Equations for Coefficients

After expressing the rational expression as a sum of partial fractions with unknown coefficients, you multiply both sides by the common denominator and equate coefficients of corresponding powers of x. Solving the resulting system of equations determines the values of these coefficients.