An elliptic inclusion with prescribed polynomial eigenstrains in an infinite Kirchhoff plate is analyzed. The integral type general solutions for the in-plane and out-of-plane displacements on the mid-plane of the plate were derived. The integrals were simplified by using Green's function for the Kirchhoff plate. The integrals could be explicitly expressed by calculating two potential functions defined in this work. After some manipulation of Ferrers and Dyson's formula related to the integration of the harmonic potential for the three-dimensional ellipsoid, we evaluated the potential functions, which can be algebraically expressed by the I-integrals. The results were applied to the analysts of the thermal stress for an inclusion with non uniform temperature distribution that might be approximated by a polynomial. For mathematical convenience, we consider an inclusion with a linear temperature distribution. The expressions for the displacements were decomposed in order to separately investigate the effects of the constant and the first-order term of the temperature distribution. The elastic fields caused by an elliptic inhomogeneity with polynomial eigenstrains, which is called the inhomogeneous inclusion, were also determined by the equivalent eigenstrain method.