HMMT 2017 A1

Let $Q(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n}$ be a polynomial with integer coefficients, and $0 \leq a_{i}<3$ for all $0 \leq i \leq n$. Given that $Q(\sqrt{3})=20+17 \sqrt{3}$, compute $Q(2)$.