Write the polynomial \( f(x) \) that meets the given conditions. Answers may vary. Degree 3 polynomial with zeros of \( -1,2 i \), and \( -2 i \). \( f(x)= \)

To construct a degree 3 polynomial with the given zeros of \(-1\), \(2i\), and \(-2i\), we can start by using the fact that the polynomial can be expressed as \( f(x) = k(x + 1)(x - 2i)(x + 2i) \). The term \((x - 2i)(x + 2i)\) simplifies to \(x^2 + 4\) (using the difference of squares). Thus, we can now write the polynomial: \[ f(x) = k(x + 1)(x^2 + 4) \] If we take \(k = 1\), the polynomial becomes: \[ f(x) = (x + 1)(x^2 + 4) = x^3 + 4x + x^2 + 4 \] Combining like terms leads to: \[ f(x) = x^3 + x^2 + 4x + 4 \] So, the polynomial is: \[ f(x) = x^3 + x^2 + 4x + 4 \]