Lucas-Gauss Theorem

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The Lucas–Gauss theorem states that the convex hull of the roots of any nonconstant complex polynomial contains the roots of its derivative. In this Demonstration there are eight locators which represent the eight roots of the polynomial

f(z)=(z-

z

1

)(z-

z

2

)⋯(z-

z

8

)

. The convex hull of these roots is shown. Within, you see the roots of the derivative of

f

(and those of the higher derivatives). Applying the theorem to each derivative, you ultimately see a nested sequence of polygons: the convex hull of the roots of

f

, which contains the convex hull of the roots of

f'

, and so on. The relationship of the roots of a polynomial to the roots of its derivatives is complex, but is easily explored with this Demonstration.