# Find the polynomial of the smallest degree which has certain zeros

Find the polynomial $f(x)$ with real coefficients of the smallest possible degree for which $i$ and $1+i$ are zeros and in which the coefficient of the highest power is 1. Also do this for the polynomial $g(x)$ with complex coefficients.

So we know that $(x-i)$ and $(x-1-i)$ are factors in the polynomial, so unless I'm missing something we have $g(x) = (x-i)(x-1-i) = x^2-x+i(1-2x)-1.$

However, I don't know how to find the polynomial with real coefficients. Can I get some hints?

Hint: If the polynomial has real coefficients and a complex root $z_0$ then the conjugate $\overline{z_0}$ must also be a root.
• So it's $f(x) = (x-i) \times (x+i) \times (x-1-i) \times (x-1+i)$? Ah, this has real coefficients. – Ewoud Sep 20 '16 at 13:40