  The Exact Factorization framework is extended and utilized to introduce the
electronic-states of correlated electron-photon systems. The formal definitions
of an exact scalar potential and an exact vector potential that account for the
electron-photon correlation are given. Inclusion of these potentials to the
Hamiltonian of the uncoupled electronic system leads to a purely electronic
Schr\"odinger equation that uniquely determines the electronic states of the
complete electron-photon system. For a one-dimensional asymmetric double-well
potential coupled to a single photon mode with resonance frequency, we
investigate the features of the exact scalar potential. In particular, we
discuss the significance of the step-and-peak structure of the exact scalar
potential in describing the phenomena of photon-assisted delocalization and
polaritonic squeezing of the electronic excited-states. In addition, we develop
an analytical approximation for the scalar potential and demonstrate how the
step-and-peak features of the exact scalar potential are captured by the
proposed analytical expression.
