A Polynomial Black Box

Algebra Level 3

The polynomial $p(x)$ is of degree 2017 and has non-negative integer coefficients which you don't know.

If you input a value like $x= x_0,$ the computer will output the value $p(x_0)$ at a cost of $1.

If you want to determine all 2018 coefficients of $p(x)$ at a minimum cost using only positive integer inputs, what is your cost in dollars?