In this paper, by using methods of $D$-companion matrix, we reprove a generalization of the Guass-Lucas theorem and 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：本文利用$D$-伴随矩阵的方法，重新证明了高斯-卢卡斯定理的一个推广形式，并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凸组合的零点集合与原多项式零点的闭凸包重合，并确定了不完全多项式凸组合的零点位置。