Question

If a polynomial function with rational coefficients has the zeros 2-2i, 1-2i, 1+2i, what are the additional zeros?

Answer 1

At least `2+2i`

Explanation:

A polynomial function with rational coefficients always has pair of complex conjugate numbers as it's zeros.

As the polynomial has the zeros `2-2i, 1-2i, 1+2i`, of which last two are pair of complex conjugate numbers, but first zero i.e. `2-2i` does not have its conjugate `2+2i` as zero. Hence, the polynomial must have atleast `2+2i` as additional zero.

There could be more additional zeros, but they would have to be either rational numbers or pair of complex conjugate numbers.