In this paper, we provide a simple proof of a generalization of the Gauss-Lucas theorem. By using methods of D-companion matrix, we get the majorization relationship between the zeros of convex combinations of incomplete polynomials and an origin polynomial. Moreover, we prove that the set of all zeros of all convex combinations of incomplete polynomials coincides with the closed convex hull of zeros of the original polynomial. The location of zeros of convex combinations of incomplete polynomials is determined.

翻译：本文给出了高斯-卢卡斯定理推广的一个简洁证明。通过利用$D$-伴随矩阵方法，我们得到了不完全多项式的凸组合的零点与原多项式零点之间的优超关系。此外，我们证明了所有不完全多项式的所有凸组合的零点集与原多项式零点的闭凸包重合。不完全多项式凸组合的零点位置得以确定。