Determine positive integers A, B, C, D, E and F such that 12x-13 D Ex-F 6x²-13-5 A= C- D= E Bx+C +
Determine positive integers A, B, C, D, E and F such that 12x-13 D Ex-F 6x²-13-5 A= C- D= E Bx+C +
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Title: Solving Rational Equations with Partial Fraction Decomposition**
**Determine Positive Integers \( A, B, C, D, E, \) and \( F \) such that:**
\[
\frac{12x - 13}{6x^2 - 13x - 5} = \frac{A}{Bx + C} + \frac{D}{Ex - F}
\]
**Objective:**
Find the positive integer values for \( A, B, C, D, E, \) and \( F \) that satisfy the equation above.
**Solution Steps:**
1. **Factor the Denominator:** Start by factoring the denominator \( 6x^2 - 13x - 5 \) to find possible linear factors. These factors will help form the denominators in the partial fractions on the right side of the equation.
2. **Decompose into Partial Fractions:** Once the denominator is factored, express the left-hand side as a sum of two separate fractions, each with linear denominators as shown.
3. **Determine Coefficients:** Equalize the numerators for both sides and solve for the unknowns \( A, B, C, D, E, \) and \( F \) by matching coefficients or by plugging in convenient values of \( x \) to generate a system of linear equations.
**Input Fields:**
- \( A = \)
- \( B = \)
- \( C = \)
- \( D = \)
- \( E = \)
- \( F = \)
This method of decomposition will allow for the simplification and easier integration of rational expressions in calculus, or for solving other algebraic problems involving polynomial divisions.
---
**Graph/Diagram Explanation:**
There are no graphs or diagrams present in this image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3632b9ca-0e22-48c8-943d-c9e08fc0f04c%2F6b626018-5542-4102-88da-e21a159c2b99%2Fy3g2bg_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Solving Rational Equations with Partial Fraction Decomposition**
**Determine Positive Integers \( A, B, C, D, E, \) and \( F \) such that:**
\[
\frac{12x - 13}{6x^2 - 13x - 5} = \frac{A}{Bx + C} + \frac{D}{Ex - F}
\]
**Objective:**
Find the positive integer values for \( A, B, C, D, E, \) and \( F \) that satisfy the equation above.
**Solution Steps:**
1. **Factor the Denominator:** Start by factoring the denominator \( 6x^2 - 13x - 5 \) to find possible linear factors. These factors will help form the denominators in the partial fractions on the right side of the equation.
2. **Decompose into Partial Fractions:** Once the denominator is factored, express the left-hand side as a sum of two separate fractions, each with linear denominators as shown.
3. **Determine Coefficients:** Equalize the numerators for both sides and solve for the unknowns \( A, B, C, D, E, \) and \( F \) by matching coefficients or by plugging in convenient values of \( x \) to generate a system of linear equations.
**Input Fields:**
- \( A = \)
- \( B = \)
- \( C = \)
- \( D = \)
- \( E = \)
- \( F = \)
This method of decomposition will allow for the simplification and easier integration of rational expressions in calculus, or for solving other algebraic problems involving polynomial divisions.
---
**Graph/Diagram Explanation:**
There are no graphs or diagrams present in this image.
Expert Solution
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