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?
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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![**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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