  In this paper we present the techniques for computing cosmological bounces in
polynomial $f(R)$ theories, whose order is more than two, for spatially flat
FLRW spacetime. In these cases the conformally connected Einstein frame shows
up multiple scalar potentials predicting various possibilities of cosmological
evolution in the Jordan frame where the $f(R)$ theory lives. We present a
reasonable way in which one can associate the various possible potentials in
the Einstein frame, for cubic $f(R)$ gravity, to the cosmological development
in the Jordan frame. The issue concerning the energy conditions in $f(R)$
theories is presented. We also point out the very important relationships
between the conformal transformations connecting the Jordan frame and the
Einstein frame and the various instabilities of $f(R)$ theory. All the
calculations are done for cubic $f(R)$ gravity but we hope the results are
sufficiently general for higher order polynomial gravity.
