Select all of the roots/zeros of \( f(x)=x^{4}+2 x^{3}-6 x^{2}+4 x-16 \) select 3 correct answer(s) \( \square^{-4} \) \( \square^{2} \) \( \pm i \sqrt{2} \) \( \square^{8} \) \( \square^{-2} \)

To find the roots of the polynomial \( f(x) = x^{4} + 2x^{3} - 6x^{2} + 4x - 16 \), we can use several methods like synthetic division, factoring, or numerical methods. After analyzing and evaluating the polynomial, one important aspect is that the roots can often be complex numbers or even irrational. In this case, through calculation (specifically polynomial division or numerical approximation), we discover that \( \pm i \sqrt{2} \) are indeed valid roots. Additionally, you can check \( -4 \) and \( 2 \) and find that they are also zeros of the function. So the correct answers among the options given would be: 1. \( -4 \) 2. \( 2 \) 3. \( \pm i \sqrt{2} \)