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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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![### Problem Statement
Resolve the rational expression
\[ \frac{2x + 3}{3x^2 + 14x - 5} \]
into partial fractions.
### Explanation
Partial fraction decomposition is a technique used to express a rational function as the sum of simpler rational expressions. Here's a detailed, step-by-step guide to solving this problem:
1. **Factorize the Denominator**: First, factorize the quadratic polynomial in the denominator.
\[ 3x^2 + 14x - 5 \]
2. **Set Up Partial Fractions**: Once the denominator is factorized, set up the partial fractions with unknown coefficients.
3. **Solve for Coefficients**: Equate the numerators after expressing them with a common denominator and solve for the unknown coefficients.
After following these steps, the original complex rational expression can be written as the sum of simpler fractions, which are easier to integrate or differentiate.
This process is fundamental in algebra and calculus, particularly in simplifying integrals and solving differential equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc7f24697-548c-411c-9619-a039c4f409b9%2F25bea489-79b9-42f7-9580-80e6c0e3cd2a%2F2qggng8.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Resolve the rational expression
\[ \frac{2x + 3}{3x^2 + 14x - 5} \]
into partial fractions.
### Explanation
Partial fraction decomposition is a technique used to express a rational function as the sum of simpler rational expressions. Here's a detailed, step-by-step guide to solving this problem:
1. **Factorize the Denominator**: First, factorize the quadratic polynomial in the denominator.
\[ 3x^2 + 14x - 5 \]
2. **Set Up Partial Fractions**: Once the denominator is factorized, set up the partial fractions with unknown coefficients.
3. **Solve for Coefficients**: Equate the numerators after expressing them with a common denominator and solve for the unknown coefficients.
After following these steps, the original complex rational expression can be written as the sum of simpler fractions, which are easier to integrate or differentiate.
This process is fundamental in algebra and calculus, particularly in simplifying integrals and solving differential equations.
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