Suppose that R (x) is a polynomial of degree 7 whose coefficients are real numbers. Also, suppose that R(x) has the following zeros. -4-5i, 5-i, -4i Answer the following. (a) Find another zero of R (x). (b) What is the maximum number of real zeros that R(x) can have? (c) What is the maximum number of nonreal zeros that R(x) can have?
Suppose that R (x) is a polynomial of degree 7 whose coefficients are real numbers. Also, suppose that R(x) has the following zeros. -4-5i, 5-i, -4i Answer the following. (a) Find another zero of R (x). (b) What is the maximum number of real zeros that R(x) can have? (c) What is the maximum number of nonreal zeros that R(x) can have?
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![**Polynomial Zeros Analysis**
Suppose that \( R(x) \) is a polynomial of degree 7 whose coefficients are real numbers.
Also, suppose that \( R(x) \) has the following zeros:
\[
-4 - 5i, \quad 5 - i, \quad -4i
\]
**Answer the following:**
(a) **Find another zero of \( R(x) \).**
(b) **What is the maximum number of real zeros that \( R(x) \) can have?**
(c) **What is the maximum number of nonreal zeros that \( R(x) \) can have?**
---
**Explanations:**
1. Since the coefficients of the polynomial are real numbers, complex zeros must occur in conjugate pairs. Therefore, for each given complex zero, its conjugate is also a zero of the polynomial.
2. **Zeros to Discover**:
- Conjugate pairs for given zeros:
- \(-4 - 5i\) implies \(-4 + 5i\)
- \(5 - i\) implies \(5 + i\)
- \(-4i\) implies \(4i\)
3. **Maximum Number of Real Zeros**:
- Since the polynomial is of degree 7, it can have up to 7 real zeros.
4. **Maximum Number of Nonreal Zeros**:
- Given the maximum degree of 7 and the requirement for conjugate pairs, at most 6 can be nonreal (3 pairs of nonreal complex conjugates), leaving at least 1 real zero.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90b823f6-a59b-43f1-924e-a48b4e93719a%2F16358e19-50f0-4021-a07e-7bb158c2eec3%2Fet5gsk7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Polynomial Zeros Analysis**
Suppose that \( R(x) \) is a polynomial of degree 7 whose coefficients are real numbers.
Also, suppose that \( R(x) \) has the following zeros:
\[
-4 - 5i, \quad 5 - i, \quad -4i
\]
**Answer the following:**
(a) **Find another zero of \( R(x) \).**
(b) **What is the maximum number of real zeros that \( R(x) \) can have?**
(c) **What is the maximum number of nonreal zeros that \( R(x) \) can have?**
---
**Explanations:**
1. Since the coefficients of the polynomial are real numbers, complex zeros must occur in conjugate pairs. Therefore, for each given complex zero, its conjugate is also a zero of the polynomial.
2. **Zeros to Discover**:
- Conjugate pairs for given zeros:
- \(-4 - 5i\) implies \(-4 + 5i\)
- \(5 - i\) implies \(5 + i\)
- \(-4i\) implies \(4i\)
3. **Maximum Number of Real Zeros**:
- Since the polynomial is of degree 7, it can have up to 7 real zeros.
4. **Maximum Number of Nonreal Zeros**:
- Given the maximum degree of 7 and the requirement for conjugate pairs, at most 6 can be nonreal (3 pairs of nonreal complex conjugates), leaving at least 1 real zero.
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