a) 1 (x-1)(x+2)(x+4)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![### Partial Fraction Decomposition
**Problem Statement:**
**2. Find the partial fraction representation of the following:**
[The remaining part of the problem is not visible in the image]
**Explanation:**
Partial fraction decomposition is a technique used to express a rational function as the sum of simpler fractions. This is particularly useful in integration, where breaking down complex fractions can simplify the integration process.
Here's the general method for solving such problems:
1. **Factor the denominator**: Break down the denominator into a product of irreducible factors.
2. **Set up the partial fraction form**: Write the structure of partial fractions based on the factors in the denominator.
3. **Multiply through by the common denominator**: This will help in equating numerator coefficients.
4. **Solve the resulting system of equations**: Determine the values of the coefficients in the partial fractions.
**Example:**
Let's say we need to find the partial fraction decomposition of \(\frac{5x + 6}{(x-1)(x+2)}\).
1. **Factor the denominator**: The denominator is already factored as \((x-1)(x+2)\).
2. **Set up the partial fraction form**:
\[
\frac{5x+6}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}
\]
3. **Multiply through by the common denominator \((x-1)(x+2)\)**:
\[
5x + 6 = A(x+2) + B(x-1)
\]
4. **Solve the resulting system of equations**:
Collect like terms and solve for \(A\) and \(B\) by comparing coefficients.
Hope this helps in understanding how to approach partial fraction decomposition. For the specific problem you're working on, you'd follow a similar approach.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45088ba1-a1a3-4a96-b462-54f2c0a5a29e%2F91317812-9df1-4194-81d5-91e392002a6a%2Fcs43b7a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Partial Fraction Decomposition
**Problem Statement:**
**2. Find the partial fraction representation of the following:**
[The remaining part of the problem is not visible in the image]
**Explanation:**
Partial fraction decomposition is a technique used to express a rational function as the sum of simpler fractions. This is particularly useful in integration, where breaking down complex fractions can simplify the integration process.
Here's the general method for solving such problems:
1. **Factor the denominator**: Break down the denominator into a product of irreducible factors.
2. **Set up the partial fraction form**: Write the structure of partial fractions based on the factors in the denominator.
3. **Multiply through by the common denominator**: This will help in equating numerator coefficients.
4. **Solve the resulting system of equations**: Determine the values of the coefficients in the partial fractions.
**Example:**
Let's say we need to find the partial fraction decomposition of \(\frac{5x + 6}{(x-1)(x+2)}\).
1. **Factor the denominator**: The denominator is already factored as \((x-1)(x+2)\).
2. **Set up the partial fraction form**:
\[
\frac{5x+6}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}
\]
3. **Multiply through by the common denominator \((x-1)(x+2)\)**:
\[
5x + 6 = A(x+2) + B(x-1)
\]
4. **Solve the resulting system of equations**:
Collect like terms and solve for \(A\) and \(B\) by comparing coefficients.
Hope this helps in understanding how to approach partial fraction decomposition. For the specific problem you're working on, you'd follow a similar approach.
![**Example Problem**
**a)** \[\dfrac{1}{(x-1)(x+2)(x+4)}\]
This equation represents a rational function where the numerator is 1 and the denominator is a product of three linear factors: \((x - 1)\), \((x + 2)\), and \((x + 4)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45088ba1-a1a3-4a96-b462-54f2c0a5a29e%2F91317812-9df1-4194-81d5-91e392002a6a%2Fbqvy5o7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Example Problem**
**a)** \[\dfrac{1}{(x-1)(x+2)(x+4)}\]
This equation represents a rational function where the numerator is 1 and the denominator is a product of three linear factors: \((x - 1)\), \((x + 2)\), and \((x + 4)\).
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