# How do you write the simplest polynomial of least degree with zeros i, and 2i?

May 24, 2017

Depending on what is meant by "simplest", ${x}^{2} - 3 i x - 2$ or ${x}^{4} + 5 {x}^{2} + 4$

#### Explanation:

For the least degree polynomial (with complex coefficients) use

$\left(x - i\right) \left(x - 2 i\right) = {x}^{2} - 3 i x - 2$.

If we need real coefficients, then, since $i$ is a zero, its conjugate $- i$ must also be a zero. And similarly for $2 i$ and $- 2 i$.
In this case the polynomial is

$\left(x - i\right) \left(x + i\right) \left(x - 2 i\right) \left(x + 2 i\right) = \left({x}^{2} + 1\right) \left({x}^{2} + 4\right) = {x}^{4} + 5 {x}^{2} + 4$