Question

find the polynomial with real coefficients of the smallest possible degree where i and 3+i are zeros and the coefficient of the highest power is 1

Answer 1

\begin{eqnarray*}(x-i)(x+i)(x-(3+i))(x-(3-i))&=&\left(x^2 + 1\right)\left((x - 3)^2 + 1\right) \\&=& x^4 -6x^3+11x^2-6x+10.\end{eqnarray*}