  Let $p$ be a polynomial in one variable whose roots either all have
multiplicity more than 1 or all have multiplicity exactly 1. It is shown that
the universal $C^*$-algebra of a relation $p(x)=0$, $\|x\| \le 1$ is
semiprojective. In the case of all roots multiple it is shown that the
universal $C^*$-algebra is also residually finite-dimensional. Applications to
polynomially compact operators are given.
