When doubling the Newton step for the computation of the largest zero of a real polynomial with all real zeros, a classical result shows that the iterates never overshoot the largest zero of the derivative of the polynomial. Here we show that when the Newton step is extended by a factor $\theta$ with $1 < \theta < 2$, the iterates cannot overshoot the zero of a different function. When $\theta=2$, our result reduces to the one for the double-step case. An analogous property exists for the smallest zero.