Which of the following cannot be the number of nonreal zeros of a polynomial of degree 5 with real coefficients?
A. 2
B. 0
C. 3
D. 4
E. None of the above
To determine the number of nonreal zeros of a polynomial with real coefficients, we can use the Fundamental Theorem of Algebra, which states that a polynomial of degree n will have exactly n complex zeros (counting multiplicity).
Since the polynomial in question is of degree 5, it can have up to 5 nonreal zeros.
Let's examine the answer choices one by one:
A. 2: If the polynomial had 2 nonreal zeros, the remaining 3 zeros would have to be real. This is possible, so having 2 nonreal zeros is a valid option.
B. 0: If the polynomial had 0 nonreal zeros, then all of its zeros would have to be real. This is also possible, so having 0 nonreal zeros is a valid option.
C. 3: If the polynomial had 3 nonreal zeros, then the remaining 2 zeros would have to be real. This is possible, so having 3 nonreal zeros is a valid option.
D. 4: If the polynomial had 4 nonreal zeros, then the remaining 1 zero would have to be real. This is possible, so having 4 nonreal zeros is a valid option.
E. None of the above: Since we have shown that each of the given options is a valid possibility, the answer is none of the above.
Therefore, the answer is E. None of the above.
To determine the number of nonreal zeros of a polynomial of degree 5 with real coefficients, we can use the Fundamental Theorem of Algebra. According to this theorem, a polynomial of degree n has exactly n complex zeros, including both real and nonreal zeros.
Since the degree of the polynomial is 5, we are looking for the number of nonreal zeros, which would be less than or equal to 5.
Let's analyze the given options one by one.
A. 2: It is possible to have 2 nonreal zeros for a polynomial of degree 5. Therefore, this option can be the number of nonreal zeros.
B. 0: If a polynomial of degree 5 has all real zeros, it means there are no nonreal zeros. Therefore, this option cannot be the number of nonreal zeros.
C. 3: It is possible to have 3 nonreal zeros for a polynomial of degree 5. Therefore, this option can be the number of nonreal zeros.
D. 4: It is possible to have 4 nonreal zeros for a polynomial of degree 5. Therefore, this option can be the number of nonreal zeros.
E. None of the above: This option is invalid as we have determined that options A, C, and D are possible numbers of nonreal zeros.
Based on our analysis, option B, 0, cannot be the number of nonreal zeros of a polynomial of degree 5 with real coefficients. Therefore, the correct answer is B.