Question 12 (1 point) Write a polynomial function of least degree with integral coefficients that has the given zeros: \( \begin{array}{ll}-1,2 i \\ \text { a b } & f(x)=x^{3}-x^{2}+4 x-4 \\ \text { c } & f(x)=-x^{3}+x^{2}-4 x+4 \\ \text { d } & f(x)=2 x^{3}+x^{2}+4 x+4 \\ \text { d } x^{3}+x^{2}+4 x+1\end{array} \)

To create a polynomial function with the given zeros of -1 and 2i, you should remember that complex zeros come in conjugate pairs. Thus, the zeros are -1, 2i, and -2i. You can express the polynomial as \( f(x) = (x + 1)(x - 2i)(x + 2i) \). The factors containing the complex zeros can be simplified into a quadratic: \( (x - 2i)(x + 2i) = x^2 + 4 \). Therefore, the polynomial becomes \( f(x) = (x + 1)(x^2 + 4) = x^3 + x^2 + 4x + 4 \). If you find yourself confused, a good tip is to always pair up complex conjugates when forming polynomials. Those little pairs help you keep everything neat and avoid missing those crucial terms that make or break your function. Also, ensure that your polynomial remains at its core a product of linear factors bound in conjugate love with integer coefficients!