Question 571790
If the coefficients are rational numbers, irrational roots have to appear in conjugate pairs, so -3i must also be a root.
A polynomial of degree 3, with 1, 3i, and -3i as roots has to be equal to
{{{K(x-1)(x-3i)(x+3i)}}} with some rational non-zero number {{{K}}}, and all polynomials of such form are of degree 3, and have 1, 3i, and -3i as roots.
So there are infinite such polynomials for the answer, but the simplest, with {{{K=1}}} is
{{{(x-1)(x-3i)(x+3i)=(x-1)(x^2+9)=x^3-x^2+9x-9}}}