Complex Roots of a Polynomial and Its Derivative

order of polynomial

3

derivative

1

domain mesh
range mesh
expand view

-1+

3

z

3

2

z

For a polynomial

p(z)

, the roots of

p'(z)

(red) always lie within the convex hull of the roots of

p(z)

(cyan). Drag the locators, which are the roots of

p(z)

scaled up.

The red curve is the image of the unit circle transformed by the derivative of the polynomial. Use the checkbox "range mesh" to see the image of the grid ("domain mesh"), again under the map

p'(z)

.