Question 6 (1 point) State the number of complex zeros and the possible number of real and imaginary zeros for each functic \( f(x)=27 x^{9}+8 x^{6}-27 x^{3}-8 \) a \# of complex zeros: 17 Possible \# of real zeros: \( 9,7,5,3 \), or 1 Possible \# of imaginary zero b \# of complex zeros: 9 Possible \# of real zeros: 5,3 , or 1 Possible \# of imaginary zeros: 8,6 c \# of complex zeros: 9 Possible \# of real zeros: \( 9,7,5,3 \), or 1 Possible \# of imaginary zeros d \# of complex zeros: 17 Possible \# of real zeros: \( 9,7,5,3 \), or 1 Possible \# of imaginary zero

To determine the number of complex zeros and the possible number of real and imaginary zeros for the function \[ f(x) = 27x^9 + 8x^6 - 27x^3 - 8, \] we can use the following principles: 1. **Fundamental Theorem of Algebra**: This theorem states that a polynomial of degree \( n \) has exactly \( n \) complex zeros (counting multiplicities). 2. **Real and Imaginary Zeros**: The number of real zeros can be odd or even, and the imaginary zeros will always occur in conjugate pairs. ### Step 1: Determine the Degree of the Polynomial The degree of the polynomial \( f(x) \) is 9 (the highest power of \( x \)). Therefore, according to the Fundamental Theorem of Algebra, the total number of complex zeros is 9. ### Step 2: Analyze Possible Real Zeros The possible number of real zeros can be determined using Descartes' Rule of Signs, which states that the number of positive real zeros is equal to the number of sign changes in \( f(x) \) when evaluated at positive values of \( x \), and the number of negative real zeros is equal to the number of sign changes in \( f(-x) \). #### For \( f(x) \): - The coefficients are \( 27, 8, -27, -8 \). - The sign changes are: \( +27 \) to \( +8 \) (no change), \( +8 \) to \( -27 \) (1 change), \( -27 \) to \( -8 \) (no change). - Total sign changes = 1. Thus, there is 1 positive real zero. #### For \( f(-x) \): \[ f(-x) = 27(-x)^9 + 8(-x)^6 - 27(-x)^3 - 8 = -27x^9 + 8x^6 + 27x^3 - 8. \] - The coefficients are \( -27, 8, 27, -8 \). - The sign changes are: \( -27 \) to \( +8 \) (1 change), \( +8 \) to \( +27 \) (no change), \( +27 \) to \( -8 \) (1 change). - Total sign changes = 2. Thus, there can be 2 or 0 negative real zeros. ### Step 3: Summarize the Findings - Total complex zeros: 9. - Possible real zeros: 1 (positive) + 2 (negative) = 3 or 1. - Since the total number of zeros is 9, the remaining zeros must be imaginary. ### Conclusion - **Number of complex zeros**: 9 - **Possible number of real zeros**: 3, 1 - **Possible number of imaginary zeros**: 6, 8 (since the total must add up to 9). Thus, the correct answer is: **b**: - Number of complex zeros: 9 - Possible number of real zeros: 5, 3, or 1 - Possible number of imaginary zeros: 8, 6.