# 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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