Question 15
Recall that the conjugate of the complex number $w = a + bi,$ where $a$ and $b$ are real numbers and $i = \sqrt{-1},$ is the complex number $\overline{w} = a - bi.$ For any complex number $z,$ let $f(z) = 4i\overline{z}.$ The polynomial $$
P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1
$$ has four complex roots: $z_1,$ $z_2,$ $z_3,$ and $z_4.$ Let $$
\begin{aligned} &Q(z) = z^4 + Az^3 + Bz^2 \\ &\quad {}+ Cz + D \end{aligned}
$$ be the polynomial whose roots are $f(z_1),$ $f(z_2),$ $f(z_3),$ and $f(z_4),$ where the coefficients $A,$ $B,$ $C,$ and $D$ are complex numbers. What is $B + D?$
P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1
$$ has four complex roots: $z_1,$ $z_2,$ $z_3,$ and $z_4.$ Let $$
\begin{aligned} &Q(z) = z^4 + Az^3 + Bz^2 \\ &\quad {}+ Cz + D \end{aligned}
$$ be the polynomial whose roots are $f(z_1),$ $f(z_2),$ $f(z_3),$ and $f(z_4),$ where the coefficients $A,$ $B,$ $C,$ and $D$ are complex numbers. What is $B + D?$