Suppose that \( R(x) \) is a polynomial of degree 8 whose coefficients are real numbers. Also, suppose that \( R(x) \) has the following zeros. Answer the following. (a) Find another zero of \( R(x) \). \( \square \) (b) What is the maximum number of real zeros that \( R(x) \) can have? \( \square \) (c) What is the maximum number of nonreal zeros that \( R(x) \) can have? (

To solve the problem, we need to analyze the properties of the polynomial \( R(x) \) based on the information provided about its zeros. ### Given Information: - \( R(x) \) is a polynomial of degree 8. - The coefficients of \( R(x) \) are real numbers. ### (a) Find another zero of \( R(x) \). Since the coefficients of \( R(x) \) are real, any nonreal zeros must occur in conjugate pairs. If we know one zero is nonreal, say \( a + bi \) (where \( a \) and \( b \) are real numbers and \( b \neq 0 \)), then its conjugate \( a - bi \) must also be a zero of \( R(x) \). If the problem provides specific zeros, we can identify the conjugate. However, without specific zeros given, we can only state that if there is a nonreal zero \( z \), then \( \overline{z} \) is also a zero. **Answer:** If \( z \) is a nonreal zero, then \( \overline{z} \) is another zero. ### (b) What is the maximum number of real zeros that \( R(x) \) can have? The maximum number of real zeros a polynomial can have is equal to its degree. Since \( R(x) \) is of degree 8, the maximum number of real zeros it can have is 8. **Answer:** 8 ### (c) What is the maximum number of nonreal zeros that \( R(x) \) can have? The total number of zeros of a polynomial is equal to its degree, which is 8 in this case. If all zeros are real, there are 0 nonreal zeros. If there are nonreal zeros, they must come in pairs (due to the conjugate root theorem). Thus, the maximum number of nonreal zeros occurs when all zeros are nonreal, which means we can have 8 nonreal zeros if they are all in pairs. However, since they must be in pairs, the maximum number of nonreal zeros is 8, but this would imply that there are no real zeros. **Answer:** 8 ### Summary of Answers: (a) If \( z \) is a nonreal zero, then \( \overline{z} \) is another zero. (b) 8 (c) 8