Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics

New exactly solvable rationally-extended radial oscillator and Scarf I potentials are generated by using a constructive supersymmetric quantum mechanical method based on a reparametrization of the corresponding conventional superpotential and on the addition of an extra rational contribution expressed in terms of some polynomial $g$. The cases where $g$ is linear or quadratic are considered. In the former, the extended potentials are strictly isospectral to the conventional ones with reparametrized couplings and are shape invariant. In the latter, there appears a variety of extended potentials, some with the same characteristics as the previous ones and others with an extra bound state below the conventional potential spectrum. Furthermore, the wavefunctions of the extended potentials are constructed. In the linear case, they contain $(\nu+1)$th-degree polynomials with $\nu=0,1,2,\dots$, which are shown to be $X_1$-Laguerre or $X_1$-Jacobi exceptional orthogonal polynomials. In the quadratic case, several extensions of these polynomials appear. Among them, two different kinds of $(\nu+2)$th-degree Laguerre-type polynomials and a single one of $(\nu+2)$th-degree Jacobi-type polynomials with $\nu=0,1,2,\dots$ are identified. They are candidates for the still unknown $X_2$-Laguerre and $X_2$-Jacobi exceptional orthogonal polynomials, respectively.