Question 15
Let $$
\begin{aligned} &P(x) = 24x^{24} \\ &\quad {}+ \sum_{j=1}^{23} (24 - j)\left(x^{24-j} + x^{24+j}\right). \end{aligned}
$$ Let $z_1,$ $z_2,$ $\ldots,$ $z_r$ be the distinct zeros of $P(x),$ and let $z_k^2 = a_k + b_k i$ for $k = 1,$ $2,$ $\ldots,$ $r,$ where $i = \sqrt{-1},$ and $a_k$ and $b_k$ are real numbers. Let $$
\sum_{k=1}^{r} |b_k| = m + n\sqrt{p},
$$ where $m,$ $n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m + n + p.$
\begin{aligned} &P(x) = 24x^{24} \\ &\quad {}+ \sum_{j=1}^{23} (24 - j)\left(x^{24-j} + x^{24+j}\right). \end{aligned}
$$ Let $z_1,$ $z_2,$ $\ldots,$ $z_r$ be the distinct zeros of $P(x),$ and let $z_k^2 = a_k + b_k i$ for $k = 1,$ $2,$ $\ldots,$ $r,$ where $i = \sqrt{-1},$ and $a_k$ and $b_k$ are real numbers. Let $$
\sum_{k=1}^{r} |b_k| = m + n\sqrt{p},
$$ where $m,$ $n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m + n + p.$