  We study generalizations of the classical Bernstein operators on polynomial
spaces, where instead of fixing $\mathbf{1}$ and $x$, we require that
$\mathbf{1}$ and a strictly increasing polynomial $f_1$ be fixed. Via several
examples, we exhibit the diversity of behaviours in this more general setting.
We also prove that for sufficiently large dimensions, there always exist
generalized Bernstein operators fixing $\mathbf{1}$ and $f_1$, and converging
to the identity.
