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Answer :
To determine the nature of the solutions for the equation [tex]\(x^5 + 6x^2 - 10x - 60 = 0\)[/tex], we need to analyze its roots. This is a fifth-degree polynomial, so it can have up to five roots. These roots can be either real or non-real (complex).
From the analysis of the roots, we have found one real solution, which is approximately 2.21, and four non-real solutions (complex solutions).
Given these findings, let's match this information with the options provided:
1. The equation has four non-real solutions.
- This is true because we have found four non-real (complex) solutions.
2. The equation has two real solutions and one non-real solution.
- This is not true because there is only one real solution.
3. The equation has all real solutions.
- This is not true since the majority of the solutions are non-real.
4. The equation has all non-real solutions.
- This is not true as there is one real solution.
Therefore, the correct description of the solutions for the equation is: "The equation has four non-real solutions."
From the analysis of the roots, we have found one real solution, which is approximately 2.21, and four non-real solutions (complex solutions).
Given these findings, let's match this information with the options provided:
1. The equation has four non-real solutions.
- This is true because we have found four non-real (complex) solutions.
2. The equation has two real solutions and one non-real solution.
- This is not true because there is only one real solution.
3. The equation has all real solutions.
- This is not true since the majority of the solutions are non-real.
4. The equation has all non-real solutions.
- This is not true as there is one real solution.
Therefore, the correct description of the solutions for the equation is: "The equation has four non-real solutions."
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