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This paper develops the equilibrium equations describing the flexoelectric effect in soft dielectrics under large deformations. Previous works have developed related theories using a flexoelectric coupling tensor of mixed material-spatial character. Here, we formulate the model in terms of a flexoelectric tensor completely defined in the material frame, with the same symmetries of the small-strain flexocoupling tensor and leading naturally to objective flexoelectric polarization fields. The energy potential and equilibrium equations are first expressed in terms of deformation and polarization, and then rewritten in terms of deformation and electric potential, yielding an unconstrained system of fourth order partial differential equations (PDEs). We further develop a theory of geometrically nonlinear extensible flexoelectric rods under open and closed circuit conditions, with which we examine analytically cantilever bending and buckling under mechanical and electrical actuation. Besides being a simple and explicit model pertinent to slender structures, this rod theory also allows us to test our general theory and its numerical implementation using B-splines. This numerical implementation is robust as it handles the electromechanical instabilities in soft flexoelectric materials.