Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 12 / c

Real coefficients imply that the conjugate $-i$ is also a polynomial root. Write the remaining root as $r$. ThenThe coefficient of $z$ in this product is $-2(2+r)$. That coefficient is zero in the specified polynomial, so $r=-2$. Hence the two other roots areAs a check, the full polynomial is $2(z^2-4)(z^2+1)=2z^4-6z^2-8$, corresponding to $a=0$, $b=-6$ and $c=-8$.