We investigate the backward Darboux transformations (addition of the lowest bound state) of shape-invariant potentials on the line, and classify the subclass of algebraic deformations, those for which the potential and the bound states are simple elementary functions. A countable family, m = 0, 1,.2,..., of deformations exists for each family of shape-invariant potentials. We prove that the mth deformation is exactly solvable by polynomials, meaning that it leaves invariant an infinite flag of polynomial modules P-m((m)) subset of P-m+1((m)) subset of (...) , where P-n((m)) is a codimension m subspace of <1, z,..., z(n)>. In particular, we prove that the first (m = 1) algebraic deformation of the shape-invariant class is precisely the class of operators preserving the infinite flag of exceptional monomial modules P-n((1)) = <1, z(2),..., z(n)>. By construction, these algebraically deformed Hamiltonians do not have an sl(2) hidden symmetry algebra structure.