In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants.(11x - 10)/(x − 2) (x + 1)

Verified step by step guidance

1. Step 1: Identify the form of the rational expression. The given expression is \( \frac{11x - 10}{(x - 2)(x + 1)} \).
2. Step 2: Recognize that the denominator is a product of two distinct linear factors: \((x - 2)\) and \((x + 1)\).
3. Step 3: Set up the partial fraction decomposition. For distinct linear factors, the decomposition takes the form \( \frac{A}{x - 2} + \frac{B}{x + 1} \), where \(A\) and \(B\) are constants to be determined.
4. Step 4: Write the partial fraction decomposition as \( \frac{11x - 10}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1} \).
5. Step 5: Note that solving for \(A\) and \(B\) is not required in this exercise, so the process stops here.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for performing operations like addition, subtraction, and decomposition. In this context, the expression (11x - 10)/(x - 2)(x + 1) is a rational expression that needs to be decomposed into simpler fractions.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is particularly useful for integrating rational functions or simplifying complex expressions. The goal is to break down the given rational expression into components that are easier to work with, typically involving linear factors in the denominator.

Linear Factors

Linear factors are expressions of the form (ax + b), where a and b are constants. In the context of partial fraction decomposition, recognizing the linear factors in the denominator is essential for determining the form of the decomposition. For the expression given, the factors (x - 2) and (x + 1) are linear, which influences how the partial fractions will be structured.