Q has degree 3, and zeros 0 and i. Since i is a zero, then so is -i by the Conjugate Zeros Theorem. This means that has the form Q(x)=a(x-0)(x+i)(x-i) Expand and simplify Q(x)=a(x^(3)+x) Let a=1 Q(x)=x^(3)+x Any other polynomial that satisfies the given requirements must be an integer multiple of this one. Result: Q(x)=x^(3)+x