Polynomial phobia abandoned!

Algebra Level 4

What is the minimum value of \(p(2)\) if the following 4 conditions are followed?

\(p(x)\) is a polynomial of degree \(17\).
All roots of \(p(x)\) are real.
All coefficients are positive.
The coefficient of \(x^{17}\) is 1.
The product of roots of \(p(x)\) is -1.

Image Credit: Wikimedia Septic Graph