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What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following? a. A factor of x-3 in the denominator b. A factor of (x-4)³ in the denominator c. A factor of x² + 2x + 6 in the denominator

What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following? a. A factor of x-3 in the denominator b. A factor of (x-4)³ in the denominator c. A factor of x² + 2x + 6 in the denominator

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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The image contains a question about partial fraction decomposition for educational purposes. It lists three different scenarios with factors in the denominator of a proper rational function and asks which terms should appear in the decomposition. The scenarios are as follows: a. A factor of \( x - 3 \) in the denominator. b. A factor of \( (x - 4)^3 \) in the denominator. c. A factor of \( x^2 + 2x + 6 \) in the denominator. Additionally, there is a multiple-choice question: **c.** When a proper rational function has a factor of \( x^2 + 2x + 6 \) in the denominator, which of the following should appear in its partial fraction decomposition? Choose the correct answer below. Options: - **A.** \(\frac{Ax}{x^2 + 2x + 6}\) - **B.** \(\frac{A_1}{x^2 + 2x + 6} + \frac{A_2}{x^2 + 2x + 6}\) - **C.** \(\frac{A}{x^2 + 2x + 6}\) - **D.** \(\frac{Ax + B}{x^2 + 2x + 6}\) There's no graph or diagram present in the image, only text.
Transcribed Image Text:The image contains a question about partial fraction decomposition for educational purposes. It lists three different scenarios with factors in the denominator of a proper rational function and asks which terms should appear in the decomposition. The scenarios are as follows: a. A factor of \( x - 3 \) in the denominator. b. A factor of \( (x - 4)^3 \) in the denominator. c. A factor of \( x^2 + 2x + 6 \) in the denominator. Additionally, there is a multiple-choice question: **c.** When a proper rational function has a factor of \( x^2 + 2x + 6 \) in the denominator, which of the following should appear in its partial fraction decomposition? Choose the correct answer below. Options: - **A.** \(\frac{Ax}{x^2 + 2x + 6}\) - **B.** \(\frac{A_1}{x^2 + 2x + 6} + \frac{A_2}{x^2 + 2x + 6}\) - **C.** \(\frac{A}{x^2 + 2x + 6}\) - **D.** \(\frac{Ax + B}{x^2 + 2x + 6}\) There's no graph or diagram present in the image, only text.
Title: Understanding Partial Fraction Decomposition --- **Question:** What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following? a. A factor of \(x - 3\) in the denominator b. A factor of \((x - 4)^3\) in the denominator c. A factor of \(x^2 + 2x + 6\) in the denominator --- **Problem:** **a.** When a proper rational function has a factor of \(x - 3\) in the denominator, which of the following should appear in its partial fraction decomposition? Choose the correct answer below. - **A.** \(\frac{Ax}{x - 3}\) - **B.** \(\frac{A}{x - 3}\) - **C.** \(\frac{A}{x - 3} + \frac{Ax}{x - 3}\) - **D.** \(\frac{A_1}{x - 3} + \frac{A_2}{x - 3}\) --- **Explanation:** For a proper rational function with a factor of \(x - 3\) in the denominator, the correct form in its partial fraction decomposition should represent potential simpler fractions based on the linear factor present. Each of the options provided reflects a different interpretation of how the factor \(x - 3\) should be decomposed. Choosing the most appropriate option depends on the form of the rational function expressed in simplest terms.
Transcribed Image Text:Title: Understanding Partial Fraction Decomposition --- **Question:** What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following? a. A factor of \(x - 3\) in the denominator b. A factor of \((x - 4)^3\) in the denominator c. A factor of \(x^2 + 2x + 6\) in the denominator --- **Problem:** **a.** When a proper rational function has a factor of \(x - 3\) in the denominator, which of the following should appear in its partial fraction decomposition? Choose the correct answer below. - **A.** \(\frac{Ax}{x - 3}\) - **B.** \(\frac{A}{x - 3}\) - **C.** \(\frac{A}{x - 3} + \frac{Ax}{x - 3}\) - **D.** \(\frac{A_1}{x - 3} + \frac{A_2}{x - 3}\) --- **Explanation:** For a proper rational function with a factor of \(x - 3\) in the denominator, the correct form in its partial fraction decomposition should represent potential simpler fractions based on the linear factor present. Each of the options provided reflects a different interpretation of how the factor \(x - 3\) should be decomposed. Choosing the most appropriate option depends on the form of the rational function expressed in simplest terms.
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