Question 65
Let $b(x) = x^{2} + x + 1$. The polynomial $x^{2015} + x^{2014} + \cdots + x + 1$ has a unique "base $b(x)$" representation
$$
x^{2015} + x^{2014} + \cdots + x + 1 = \sum_{k=0}^{N} a_{k}(x) b(x)^{k}
$$
where
- $N$ is a nonnegative integer;
- each "digit" $a_{k}(x)$ (for $0 \leq k \leq N$) is either the zero polynomial (i.e. $a_{k}(x) = 0$) or a nonzero polynomial of degree less than $\operatorname{deg} b = 2$; and
- the "leading digit $a_{N}(x)$" is nonzero (i.e. not the zero polynomial).
Find $a_{N}(0)$ (the "leading digit evaluated at 0").
$$
x^{2015} + x^{2014} + \cdots + x + 1 = \sum_{k=0}^{N} a_{k}(x) b(x)^{k}
$$
where
- $N$ is a nonnegative integer;
- each "digit" $a_{k}(x)$ (for $0 \leq k \leq N$) is either the zero polynomial (i.e. $a_{k}(x) = 0$) or a nonzero polynomial of degree less than $\operatorname{deg} b = 2$; and
- the "leading digit $a_{N}(x)$" is nonzero (i.e. not the zero polynomial).
Find $a_{N}(0)$ (the "leading digit evaluated at 0").