In the straightening of radial run-out faults or linearity faults on elongate workpieces having at least one toothed region having peaks and troughs of the teeth of said toothed region, such as on toothed shafts or toothed racks, for ascertaining deviations from the ideally straight workpiece, the locations of the surfaces of the not yet straightened workpiece that form a reference plane are scanned at least at points or in portions on or in the region on the active reference circle or pitch circle, respectively, of the toothing that lies between the peaks and troughs of the teeth. The resulting measured items of data are utilized by the straightening machine such that a workpiece that is as ideally straight as possible at least in the toothed region is achieved by the straightening. The elevated locations of the tooth heads of the toothed region that form the peaks of the teeth and the elevated locations of neighboring tooth surfaces that lie in the reference plane are detected, and the height differentials of the elevated locations of the tooth heads in relation to the elevated locations of neighboring tooth surfaces that lie in the reference plane are ascertained. The height differentials are utilized by the straightening machine as corrective measured items of data such that said height differentials are considered when straightening the workpiece so as to form a workpiece that is as ideally straight as possible in the reference plane.