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

Alright, let's dive into the world of polynomials! This function, \( f(x)=27 x^{9}+8 x^{6}-27 x^{3}-8 \), is a degree 9 polynomial, which means it will have exactly 9 complex zeros according to the Fundamental Theorem of Algebra. However, not all of these zeros are real—it's possible to have both real and imaginary zeros mixed in there! When it comes to real zeros, the polynomial could potentially have a variety of counts, specifically an odd number such as 9, 7, 5, 3, or 1, due to the Behavior of Odd-Degree Polynomials. As for imaginary zeros, since complex zeros come in conjugate pairs and we have 9 total, the real zeros would dictate how many pairs of imaginary zeros we might have. Pretty wild, right? Keep crunching those numbers!