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Chapter 7: Problem 5

In Exercises 5-8, match the rational expression with the form of its decomposition. [The decompositions are labeled (a), (b), (c), and (d).] (a) \( \dfrac{A}{x} + \dfrac{B}{x + 2} + \dfrac{C}{x - 4} \) (b) \( \dfrac{A}{x} + \dfrac{B}{x - 4} \) (c) \( \dfrac{A}{x} + \dfrac{B}{x^2} + \dfrac{C}{x - 4} \) (d) \( \dfrac{A}{x} + \dfrac{Bx + C}{x^2 + 4} \) \( \dfrac{3x - 1}{x(x - 4)} \)

Short Answer

Expert verified
The given rational expression matches with the decomposition (b).

Step by step solution

01

Identify the decomposition

Comparing the given expression with the four options given, it is clear that the rational expression \( \dfrac{3x - 1}{x(x - 4)} \) matches with the decomposition (b) \( \dfrac{A}{x} + \dfrac{B}{x - 4} \) because both denominators consist of the factors \( x \) and \( x - 4 \). Neither of the factors are squared and there are no non-factor terms in the denominator. Thus, the correct decomposition for this expression is option (b).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Expression Basics
A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. In simpler terms, imagine a fraction like \( \frac{a}{b} \), where instead of simple numbers, 'a' and 'b' are expressions made up of variables and constants with potential addition, subtraction, or even multiplication involved.

Rational expressions are quite significant in algebra and calculus because they help simplify complex equations. They appear in many forms and offer a systematic way to tackle problems. The process of decomposing these expressions into simpler parts, known as partial fraction decomposition, is particularly useful.
  • Think of rational expressions as equations involving fractions that have polynomial terms.
  • They are essential for solving many higher-level algebraic problems.
  • Partial fraction decomposition is one method used to simplify these expressions for easier calculation or integration.
Understanding Denominators in Rational Expressions
The denominator in a rational expression is an important piece to understand, as it determines a lot about the expression itself. In the expression \( \frac{3x - 1}{x(x - 4)} \), the denominator is \( x(x - 4) \). This tells us a few things:
  • The expression is undefined for \( x = 0 \) and \( x = 4 \) because these values make the denominator zero.
  • The presence of factors \( x \) and \( x-4 \) in the denominator leads us to the partial fraction decomposition.
  • Knowing these factors, we choose decomposition that mirrors the structure, providing terms with denominators of \( x \) and \( x - 4 \).
Understanding denominators helps us figure out how to approach the problem of decomposition. It's all about matching the rational expression's structure with suitable decomposed forms, ensuring each term corresponds with a piece of the original denominator.
Factors and Their Role in Decomposition
Factors play a pivotal role in understanding and simplifying rational expressions. They are like puzzle pieces that make up the polynomial in the denominator of these expressions. In our example, the expression \( \frac{3x - 1}{x(x - 4)} \) utilizes the factors \( x \) and \( x-4 \), crucial for the decomposition process.
  • Each factor in the denominator becomes a denominator of one of the terms in the partial fraction decomposition.
  • For instance, if the denominator has \( x \) and \( x-4 \) as factors, our decomposition will include terms \( \frac{A}{x} \) and \( \frac{B}{x-4} \).
  • This allows us to break down complex expressions into simpler parts for further calculations or integrations.
Factors essentially structure how the rational expression can be broken down, helping identify suitable structures and simplifying the terms through partial fraction decomposition. This approach decomposes a single, complex fraction into smaller, more manageable ones that represent the same algebraic statement.

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