**Exploring the Fundamental Theorem of Algebra: Identifying Complex Roots**The Fundamental Theorem of Algebra is essential in polynomial mathematics. It states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This theorem helps us determine the nature and number of roots, including complex roots, of polynomial equations.Below are several polynomial equations. Determine how many complex roots each equation has. Write your answer as a single number in the space provided. If, for example, there are twelve complex roots, type `12`.1. $ x(x^2 - 4)(x^2 + 16) = 0 $ **Number of complex roots:** 22. $ (x^2 + 4)(x + 5)^2 = 0 $ **Number of complex roots:** 23. $ x^6 - 4x^5 - 24x^2 + 10x - 3 = 0 $ **Number of complex roots:** 44. $ x^7 + 128 = 0 $ **Number of complex roots:** 65. $ (x^3 + 9)(x^2 - 4) = 0 $ **Number of complex roots:** 4*Next Steps:*- To continue with more questions, click on "NEXT QUESTION".- If you need further assistance, click on "ASK FOR HELP".- Once you're finished, click "TURN IT IN".*Note:* The equations are presented as polynomials, and the determined number of complex roots is based on their structure and coefficients. Understanding and applying the Fundamental Theorem of Algebra enables you to identify and count the complex roots effectively.

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equations have? Write your answer as a number in the space provided. For example, if there are twelv complex roots, type 12. ¤(a? – 4)(x² + 16) = 0 has 2 complex roots - (22 + 4)(x + 5)² = 0 has 2 complex roots 26 - 4x5- 24x2 + 10x-3 =0 has 4 complex roots

equations have? Write your answer as a number in the space provided. For example, if there are twelv complex roots, type 12. ¤(a? – 4)(x² + 16) = 0 has 2 complex roots - (22 + 4)(x + 5)² = 0 has 2 complex roots 26 - 4x5- 24x2 + 10x-3 =0 has 4 complex roots

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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Question

**Exploring the Fundamental Theorem of Algebra: Identifying Complex Roots**

The Fundamental Theorem of Algebra is essential in polynomial mathematics. It states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This theorem helps us determine the nature and number of roots, including complex roots, of polynomial equations.

Below are several polynomial equations. Determine how many complex roots each equation has. Write your answer as a single number in the space provided. If, for example, there are twelve complex roots, type `12`.

1. $ x(x^2 - 4)(x^2 + 16) = 0 $
**Number of complex roots:** 2

2. $ (x^2 + 4)(x + 5)^2 = 0 $
**Number of complex roots:** 2

3. $ x^6 - 4x^5 - 24x^2 + 10x - 3 = 0 $
**Number of complex roots:** 4

4. $ x^7 + 128 = 0 $
**Number of complex roots:** 6

5. $ (x^3 + 9)(x^2 - 4) = 0 $
**Number of complex roots:** 4

*Next Steps:*

- To continue with more questions, click on "NEXT QUESTION".
- If you need further assistance, click on "ASK FOR HELP".
- Once you're finished, click "TURN IT IN".

*Note:* The equations are presented as polynomials, and the determined number of complex roots is based on their structure and coefficients. Understanding and applying the Fundamental Theorem of Algebra enables you to identify and count the complex roots effectively.

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