Interpolation is a core operation that has widespread use in computer graphics. Though higher-order interpolation provides better quality, linear interpolation is often preferred due to its simplicity, performance, and hardware support. We present a unified refactoring of quadratic and cubic interpolations as standard linear interpolation plus linear interpolations of higher-order terms and show how they can be applied to regular grids and (triangular/tetrahedral) simplexes. Our formulations can provide significant reduction in computation cost in higher-order interpolations.