## Largest roots of cubic polynomials

### Problem 356

Let `a`_{n} be the largest real root of a polynomial `g`(x) = x^{3} - 2^{n}·x^{2} + `n`.

For example, `a`_{2} = 3.86619826...

Find the last eight digits of $\sum \limits_{i = 1}^{30} {\left \lfloor a_i^{987654321} \right \rfloor}$.

__ Note__: $\lfloor a \rfloor$ represents the floor function.