Question 6
For polynomial $P(x) = 1 - \frac{1}{3}x + \frac{1}{6}x^2,$ define $$
\begin{aligned} Q(x) &= P(x)P(x^3)P(x^5) \\ &\quad {}\cdot P(x^7)P(x^9) \\ &= \sum_{i=0}^{50} a_i x^i. \end{aligned}
$$ Then $\sum_{i=0}^{50} |a_i| = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
\begin{aligned} Q(x) &= P(x)P(x^3)P(x^5) \\ &\quad {}\cdot P(x^7)P(x^9) \\ &= \sum_{i=0}^{50} a_i x^i. \end{aligned}
$$ Then $\sum_{i=0}^{50} |a_i| = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$