Question 4 (1 point) State the number of complex zeros and the possible number of real and imaginary zeros for each functic \( f(x)=3 x^{3}+2 x^{2}-x \) a \( \quad \) \# of complex zeros: 5 Possible \# of real zeros: 2 or 0 Possible \# of imaginary zeros: 3 or 1 b \( \quad \) \# of complex zeros: 3 Possible \# of real zeros: 2 or 0 Possible \# of imaginary zeros: 3 or 1 d of complex zeros: 5 Possible \# of real zeros: 3 or 1 Possible \# of imaginary zeros: 2 or 0 d of complex zeros: 3 Possible \# of real zeros: 3 or 1 Possible \# of imaginary zeros: 2 or 0

The polynomial \( f(x) = 3x^3 + 2x^2 - x \) is a cubic equation, and according to the Fundamental Theorem of Algebra, it must have exactly 3 complex zeros (counting multiplicities). This means it can have various combinations of real and imaginary zeros — specifically, the possible number of real zeros can be 3, 1, or none, leading to the imaginary zeros being 0, 2, or 3 respectively. If you're looking to find those zeros, one handy approach is to try the Rational Root Theorem first! This nifty theorem tells you to test factors of the constant term over the leading coefficient for potential rational roots. Don’t forget to keep an eye out for common mistakes, like skipping over synthetic division or forgetting to consider negative roots, which might lead you to miss out on some of those real zeros hiding in plain sight!