How do you write a polynomial with zeros: 2i, sqrt2, and 2?

Mar 9, 2016

${x}^{3} - \left(2 + \sqrt{2} + 2 i\right) {x}^{2} + \left(2 \sqrt{2} + \left(2 \sqrt{2} + 4\right) i\right) x - 4 \sqrt{2} i$

Explanation:

The polynomial with zeros: $2 i , \sqrt{2}$ and $2$ is

$\left(x - \sqrt{2}\right) \left(x - 2\right) \left(x - 2 i\right)$ or

$\left({x}^{2} - \sqrt{2} x - 2 x + 2 \sqrt{2}\right) \left(x - 2 i\right)$ or

$\left({x}^{3} - \sqrt{2} {x}^{2} - 2 {x}^{2} + 2 \sqrt{2} x - 2 i {x}^{2} + 2 \sqrt{2} i x + 4 i x - 4 \sqrt{2} i\right)$ or

${x}^{3} - \left(2 + \sqrt{2} + 2 i\right) {x}^{2} + \left(2 \sqrt{2} + \left(2 \sqrt{2} + 4\right) i\right) x - 4 \sqrt{2} i$