  In the framework of polynomial Palatini cosmology, we investigate a simple
cosmological homogeneous and isotropic model with matter in the Einstein frame.
We show that in this model during cosmic evolution, it appears the early
inflation and the accelerating phase of the expansion for the late times. In
this frame we obtain the Friedmann equation with matter and dark energy in the
form of a scalar field with the potential whose form is determined in a
covariant way by the Ricci scalar of the FRW metric. The energy density of
matter and dark energy are also parametrized through the Ricci scalar. The
early inflation is obtained only for an infinitesimally small fraction of
energy density of matter. Between the matter and dark energy, there exists
interaction because the dark energy is decaying. For characterization of
inflation we calculate the slow roll parameters and the constant roll parameter
in terms of the Ricci scalar. We have found a characteristic behaviour of the
time dependence of density of dark energy on the cosmic time following the
logistic-like curve which interpolates two almost constant value phases. From
the required numbers of $N$-folds we have found a bound on model parameter.
