Prove that all three roots of $f(x)=0$ are zero. Also prove that $a+b+c=0$.

Let $x_1, x_2, x_3$ be the roots. We have $f(x) = (x - x_1)(x - x_2)(x-x_3)$.

Hence

\begin{align} 1 &= |f(i)|^2 \\ &= f(i)\overline{f(i)} \\ &= (i - x_1)(i - x_2)(i - x_3)(-i - x_1)(-i - x_2)(-i - x_3) \\ &= (x_1^2 + 1)(x_2^2 + 1)(x_3^2 + 1) \end{align}