The theorem of Gopengauz guarantees the existence of a polynomial which well approximates a function f εCq [— 1,1], while at the same time its kth derivative (k ≤ q) well approximates the kth derivative of the function, and moreover the polynomial and its derivatives respectively interpolate the function and its derivatives at ±1. With more generality, we shall prescribe that the polynomial interpolate the function at up to q+1 points near 1 and up to q + 1 points near —1. The points may coalesce, in which case one also interpolates at the coalescent point a number of derivatives one less than the multiplicity of coalescence. Aside from intrinsic theoretical interest, our results are clearly applicable in describing more precisely the error incurred in certain linear processes of simultaneous approximation, such as interpolation with added nodes near ±1. The original theorem of Gopengauz will be shown to follow as a special case.