8 S 3 4 9. 3X tex -X-7X - x +7= 0 Use Euler's method to approximate up to 0.2 accuracy the real root with the largest absolute value of the polynomial equation given above.
8 S 3 4 9. 3X tex -X-7X - x +7= 0 Use Euler's method to approximate up to 0.2 accuracy the real root with the largest absolute value of the polynomial equation given above.
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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![Title: Solving Polynomial Equations Using Euler's Method
Introduction:
In this exercise, we explore how to approximate the real root with the largest absolute value of a given polynomial equation using Euler's method, aiming for an accuracy of up to 0.2.
Polynomial Equation:
\[
3x^8 + 6x^5 - x^3 - 7x^4 - x + 7 = 0
\]
Objective:
Apply Euler’s method to approximate the real root with the largest absolute value, ensuring the result is within an accuracy of 0.2 units.
Methodology:
Euler’s method is generally used for numerical solutions of differential equations, but here, we adapt the iterative mindset to approximate the root of a polynomial. The method will involve initial approximations and adjustments until the desired accuracy is achieved.
Steps to Solve:
1. **Initial Guess**: Choose an initial approximation (could be based on graphical analysis or estimation).
2. **Iterate**: Apply the iterations using Euler’s adjustment to converge towards the accurate root.
3. **Check Accuracy**: Ensure the approximation is within the 0.2 target accuracy by evaluating the polynomial at the guessed root.
Conclusion:
By following this structured approach, we aim to identify the polynomial's most significant real root using Euler’s method, tailored to achieve precise approximation within specified limits.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F278bc67a-2bd8-419b-9abb-7528a93e8a19%2F4c59a435-01ee-4bb3-ad01-c561778361ae%2Fz8ymyqo_processed.png&w=3840&q=75)
Transcribed Image Text:Title: Solving Polynomial Equations Using Euler's Method
Introduction:
In this exercise, we explore how to approximate the real root with the largest absolute value of a given polynomial equation using Euler's method, aiming for an accuracy of up to 0.2.
Polynomial Equation:
\[
3x^8 + 6x^5 - x^3 - 7x^4 - x + 7 = 0
\]
Objective:
Apply Euler’s method to approximate the real root with the largest absolute value, ensuring the result is within an accuracy of 0.2 units.
Methodology:
Euler’s method is generally used for numerical solutions of differential equations, but here, we adapt the iterative mindset to approximate the root of a polynomial. The method will involve initial approximations and adjustments until the desired accuracy is achieved.
Steps to Solve:
1. **Initial Guess**: Choose an initial approximation (could be based on graphical analysis or estimation).
2. **Iterate**: Apply the iterations using Euler’s adjustment to converge towards the accurate root.
3. **Check Accuracy**: Ensure the approximation is within the 0.2 target accuracy by evaluating the polynomial at the guessed root.
Conclusion:
By following this structured approach, we aim to identify the polynomial's most significant real root using Euler’s method, tailored to achieve precise approximation within specified limits.
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