Consider the set $S = \{x \in \mathbb{R} \left| f(x) = 0\right. \},$ where $f$ is a *reciprocal monic polynomial with integer coefficients* (reciprocal means that the sequence of coefficients reads the same left to right or right to left; for example, the characteristic polynomials of symplectic matrices are reciprocal (conversely, every polynomial in italics (and of even degree) is the characteristic polynomial of some matrix in $Sp(2k, \mathbb{Z}).$ The question is: 

Is $S$ dense in $\mathbb{R}?$