# Roots of a random polynomial!

**Discrete Mathematics**Level 5

Let \(P(x)\) be a polynomial of degree \(d\), given by \[P(x)=c_0+c_1x+c_2x^2 +\ldots + c_dx^d.\] Each of the coefficients are chosen independently and uniformly at random from the \(p\)-element set \[\mathbb{F}_p=\{0,1,2, \ldots, p-1\},\] where \(p\) is a given prime number.

An element \(k \in \mathbb{F}_p\) is said to be a **root of the polynomial modulo \(p\)** if and only if \(P(k)\equiv 0 \pmod p\) (i.e. the prime \(p\) divides the integer \(P(k)\)).

Since the coefficients are random, so is the polynomial \(P(x)\) and so are its roots (defined strictly in the above sense). For \(p=2017\) and \(d=1008\), find the **expected number of roots** of such a random polynomial.

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