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c(x)=x°-4x+-x²+4

c(x)=x°-4x+-x²+4

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Polynomial Root Analysis For the given polynomial, perform the following tasks: - Determine the maximum possible number of positive and negative roots. - Identify all integer roots. - Find the non-integer real roots. - Ascertain the complex roots. The given polynomial is: \[ c(x) = -x^6 - 4x^4 - x^2 + 4 \] **Steps:** 1. **Maximum Possible Number of Positive Roots:** - Use Descartes' Rule of Signs to determine the maximum number of positive roots. 2. **Maximum Possible Number of Negative Roots:** - Apply Descartes' Rule of Signs to find out the maximum number of negative roots. 3. **Integer Roots:** - Use the Rational Root Theorem to list and test possible rational roots, then identify which are integers. 4. **Non-Integer Real Roots:** - Analyze the remaining roots after finding the integer ones for non-integer real roots. 5. **Complex Roots:** - After identifying all real roots, determine the complex roots, if any exist. **Details:** - **Maximum Possible Number of Positive Roots:** - \(\boxed{\text{ }\ \ \ }\) - **Maximum Possible Number of Negative Roots:** - \(\boxed{\text{ }\ \ \ }\) - **Integer Roots:** - The integer root(s) is/are: - Example: \(x=3\), \(x=-5\) - **Non-Integer Real Roots:** - The non-integer real root(s) is/are: - Example: \(x=-3.5\) - **Complex Roots:** - The complex root(s) is/are: - Example: \(x=3 + 2i\) Please proceed by using relevant mathematical software or manual calculations to fill in the exact values.
Transcribed Image Text:### Polynomial Root Analysis For the given polynomial, perform the following tasks: - Determine the maximum possible number of positive and negative roots. - Identify all integer roots. - Find the non-integer real roots. - Ascertain the complex roots. The given polynomial is: \[ c(x) = -x^6 - 4x^4 - x^2 + 4 \] **Steps:** 1. **Maximum Possible Number of Positive Roots:** - Use Descartes' Rule of Signs to determine the maximum number of positive roots. 2. **Maximum Possible Number of Negative Roots:** - Apply Descartes' Rule of Signs to find out the maximum number of negative roots. 3. **Integer Roots:** - Use the Rational Root Theorem to list and test possible rational roots, then identify which are integers. 4. **Non-Integer Real Roots:** - Analyze the remaining roots after finding the integer ones for non-integer real roots. 5. **Complex Roots:** - After identifying all real roots, determine the complex roots, if any exist. **Details:** - **Maximum Possible Number of Positive Roots:** - \(\boxed{\text{ }\ \ \ }\) - **Maximum Possible Number of Negative Roots:** - \(\boxed{\text{ }\ \ \ }\) - **Integer Roots:** - The integer root(s) is/are: - Example: \(x=3\), \(x=-5\) - **Non-Integer Real Roots:** - The non-integer real root(s) is/are: - Example: \(x=-3.5\) - **Complex Roots:** - The complex root(s) is/are: - Example: \(x=3 + 2i\) Please proceed by using relevant mathematical software or manual calculations to fill in the exact values.
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