Write out the partial fraction decomposition of x3 + 6x – 2 x4 + 6x² What is the constant multiple to the rational term with ? denominator x²? That is, x2
Write out the partial fraction decomposition of x3 + 6x – 2 x4 + 6x² What is the constant multiple to the rational term with ? denominator x²? That is, x2
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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![### Partial Fraction Decomposition Exercise
**Problem Statement:**
Write out the partial fraction decomposition of
\[ \frac{x^3 + 6x - 2}{x^4 + 6x^2} \]
What is the constant multiple to the rational term with denominator \( x^2 \)? That is,
\[ \frac{?}{x^2} \]
**Solution Box:**
---
In this problem, you're asked to decompose the given rational expression in terms of simpler fractions. The main focus is on identifying the constant multiple associated with the term having \( x^2 \) as its denominator. Partial fraction decomposition is useful for integrating rational functions, simplifying complex expressions, and solving differential equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1bc59cb9-d57c-45f6-809f-14f736729ead%2F31abfa20-1c3c-46ca-9381-f73e47ac935a%2Fuc0hs5n_processed.png&w=3840&q=75)
Transcribed Image Text:### Partial Fraction Decomposition Exercise
**Problem Statement:**
Write out the partial fraction decomposition of
\[ \frac{x^3 + 6x - 2}{x^4 + 6x^2} \]
What is the constant multiple to the rational term with denominator \( x^2 \)? That is,
\[ \frac{?}{x^2} \]
**Solution Box:**
---
In this problem, you're asked to decompose the given rational expression in terms of simpler fractions. The main focus is on identifying the constant multiple associated with the term having \( x^2 \) as its denominator. Partial fraction decomposition is useful for integrating rational functions, simplifying complex expressions, and solving differential equations.
![### Partial Fraction Decomposition
The task is to find the partial fraction decomposition of the function:
\[
\frac{x^3 + 6x - 2}{x^4 + 6x^2}
\]
Below are the possible decompositions to choose from:
- \( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^2+6} \)
- \( \frac{A}{x} + \frac{B}{x^2} + \frac{Cx+D}{x^2+6} \)
- \( \frac{A}{x^2} + \frac{Bx+C}{x^2+6} \)
- \( \frac{A}{x^2} + \frac{B}{x^2+6} \)
Select the correct decomposition based on the structure of the function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1bc59cb9-d57c-45f6-809f-14f736729ead%2F31abfa20-1c3c-46ca-9381-f73e47ac935a%2Fiaxmmcg_processed.png&w=3840&q=75)
Transcribed Image Text:### Partial Fraction Decomposition
The task is to find the partial fraction decomposition of the function:
\[
\frac{x^3 + 6x - 2}{x^4 + 6x^2}
\]
Below are the possible decompositions to choose from:
- \( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^2+6} \)
- \( \frac{A}{x} + \frac{B}{x^2} + \frac{Cx+D}{x^2+6} \)
- \( \frac{A}{x^2} + \frac{Bx+C}{x^2+6} \)
- \( \frac{A}{x^2} + \frac{B}{x^2+6} \)
Select the correct decomposition based on the structure of the function.
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