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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Write the partial fraction decomposition of the rational expression
![**Title: Partial Fraction Decomposition of Rational Expressions**
**Objective:**
Learn how to perform the partial fraction decomposition of rational expressions.
**Problem Statement:**
Write the partial fraction decomposition of the rational expression:
\[ \frac{x}{x^2 - x - 30} \]
**Explanation:**
1. **Identify the Denominator:**
The expression in the denominator is \(x^2 - x - 30\). We need to factor this quadratic expression.
2. **Factoring the Quadratic:**
- Try finding two numbers that multiply to \(-30\) and add to \(-1\) (the coefficient of \(x\)).
- These numbers are \(5\) and \(-6\).
- Therefore, the factorization is \((x - 6)(x + 5)\).
3. **Set Up Partial Fractions:**
Express the original fraction as a sum of fractions with unknown coefficients:
\[ \frac{x}{(x - 6)(x + 5)} = \frac{A}{x - 6} + \frac{B}{x + 5} \]
4. **Solve for Coefficients A and B:**
To find \(A\) and \(B\), multiply through by the original denominator to clear the fractions:
\[ x = A(x + 5) + B(x - 6) \]
5. **Expand and Collect Like Terms:**
Solve for \(A\) and \(B\) by substituting values for \(x\) to eliminate one of the variables or by comparing coefficients.
By following these steps, you'll successfully decompose the rational expression into partial fractions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb50150a-e50b-4a78-ab84-354e14945377%2F55431eb6-cc79-458a-bd50-73e82680534a%2F76vj3wh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Partial Fraction Decomposition of Rational Expressions**
**Objective:**
Learn how to perform the partial fraction decomposition of rational expressions.
**Problem Statement:**
Write the partial fraction decomposition of the rational expression:
\[ \frac{x}{x^2 - x - 30} \]
**Explanation:**
1. **Identify the Denominator:**
The expression in the denominator is \(x^2 - x - 30\). We need to factor this quadratic expression.
2. **Factoring the Quadratic:**
- Try finding two numbers that multiply to \(-30\) and add to \(-1\) (the coefficient of \(x\)).
- These numbers are \(5\) and \(-6\).
- Therefore, the factorization is \((x - 6)(x + 5)\).
3. **Set Up Partial Fractions:**
Express the original fraction as a sum of fractions with unknown coefficients:
\[ \frac{x}{(x - 6)(x + 5)} = \frac{A}{x - 6} + \frac{B}{x + 5} \]
4. **Solve for Coefficients A and B:**
To find \(A\) and \(B\), multiply through by the original denominator to clear the fractions:
\[ x = A(x + 5) + B(x - 6) \]
5. **Expand and Collect Like Terms:**
Solve for \(A\) and \(B\) by substituting values for \(x\) to eliminate one of the variables or by comparing coefficients.
By following these steps, you'll successfully decompose the rational expression into partial fractions.
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