Consider the polynomial function given by the equation:

$$ f(x) = 2x^4 - 12x^3 + 4x^2 + 8 $$

Let $r$ be a root of the equation $f(x) = 0$. Determine the value of $r$ such that:

$$ f(r) = 0 $$

Using polynomial long division, or synthetic division, determine any possible rational roots using the Rational Root Theorem, and factor the polynomial completely.

Provide the smallest positive rational root of this polynomial as your answer.