  Let L be the zero set of a nonconstant monic polynomial with complex
coefficients. In the context of constructive mathematics without countable
choice, it may not be possible to construct an element of L. In this paper we
introduce a notion of distance from a point to a subset, more general than the
usual one, that allows us to measure distances to subsets like $L$. To verify
the correctness of this notion, we show that the zero set of a polynomial
cannot be empty---a weak fundamental theorem of algebra. We also show that the
zero sets of two polynomials are a positive distance from each other if and
only if the polynomials are comaximal. Finally, the zero set of a polynomial is
used to construct a separable Riesz space, in which every element is normable,
that has no Riesz homomorphism into the real numbers.
