Question 13
The polynomial $$
\begin{aligned} P(x) &= \small (1 + x + x^2 + \cdots + x^{17})^2 \\ &\quad {}- x^{17} \end{aligned}
$$ has $34$ complex zeros of the form $z_k$ $= r_k[\cos(2\pi\alpha_k) + i\sin(2\pi\alpha_k)],$ $k = 1,$ $2,$ $3,$ $\ldots,$ $34,$ with $0 \lt \alpha_1 \le \alpha_2 \le \alpha_3$ $\le \cdots \le \alpha_{34} \lt 1$ and $r_k \gt 0.$ Given that $\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 + \alpha_5 = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$
\begin{aligned} P(x) &= \small (1 + x + x^2 + \cdots + x^{17})^2 \\ &\quad {}- x^{17} \end{aligned}
$$ has $34$ complex zeros of the form $z_k$ $= r_k[\cos(2\pi\alpha_k) + i\sin(2\pi\alpha_k)],$ $k = 1,$ $2,$ $3,$ $\ldots,$ $34,$ with $0 \lt \alpha_1 \le \alpha_2 \le \alpha_3$ $\le \cdots \le \alpha_{34} \lt 1$ and $r_k \gt 0.$ Given that $\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 + \alpha_5 = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$