Roots of the Derivatives of a Certain Real Polynomial in the Complex Plane

n

1

2

3

4

5

6

7

8

9

10

r

roots of the first n derivatives of f(z)=(z-r

n

)\),

(z+i)(z-i).

With

f(z)=

n

(z-r)

(

2

z

+1)=

n

(z-r)

(z+i)(z-i)

and

r

real,

f

is a real polynomial with three roots:

±i

and

r

. This Demonstration shows the roots of

f'(z)

,

f''(z)

, …,

(n)

f

(z)

. Each of these derivatives has at most two nonreal roots (depending on the value of the real root

r

). As

r

varies over the reals, these nonreal roots trace out ellipses in the complex plane. The nonreal roots of the first derivative lie on a circle, while those of the higher derivatives lie on successively narrower ellipses. Set the value of

n

, and slide

r

along the real line to see the roots of the derivatives move in real time.