Question

Write the simplest polynomial with the zeros 2-i, Square root 5, and -2

Answer 1

f(x)=(x^2-5)(x-(-2))((x-2+i)((x-2-i)
if coefficient are integers

Answer 2

I assume you are looking for a polynomial with real coefficients.

If that's the case, then a polynomial with real coefficients can only have complex roots if they come in complex conjugate pairs.

That means, by having complex root 2 - i, you also have root 2 + i.
In addition, you are told you also have roots \(\sqrt5\) and -2.

That means you have the following 4 factors:

\(y = (x - (2 - i))(x - (2 + i))(x - \sqrt{5})(x - (-2)) \)

\( y = (x - 2 + i)(x - 2 - i)(x - \sqrt{5})(x + 2) \)