Finding the equation to the variables with imaginary numbers

Question

The numbers are $2, -i, i$.

The answer is $x^3 - 2x^2 + x - 2 = 0$.

But I have no earthly idea how to get from the answer to the problem with imaginary numbers.

Answer 1

Hint: $(x-i)(x+i)(x-2) = 0$, multiply the left hand side all out ! You've used the following fact:

If $z$ is a zero, then $x-z$ is a factor of the polynomial being discussed. How many zeroes do you consider?

Point is you need not mention both $\pm i$ as zeroes, and only admitted one of them is a zero, then you automatically get the other due to the Fundamental Theorem of Algebra. That is if $z$ is a zero of $P(x)$ in $\mathbb{C}$, then $\overline{z}$ is also a zero.