  The symmetrized quartic polynomial oscillator is shown to admit an sl(2,$\R$)
algebraization. Some simple quasi-exactly solvable (QES) solutions are
exhibited. A new symmetrized sextic polynomial oscillator is introduced and
proved to be QES by explicitly deriving some exact, closed-form solutions by
resorting to the functional Bethe ansatz method. Such polynomial oscillators
include two categories of QES potentials: the first one containing the
well-known analytic sextic potentials as a subset, and the second one of novel
potentials with no counterpart in such a class.
