How the Roots of a Polynomial Depend on Its Constant Coefficient

polynomial

5

x

-x +(0)

This Demonstration shows how the roots (blue points) of the polynomial

p(z)=

n

z

-z+a

depend on the constant coefficient

a

, which is shown enclosed in parentheses. The polynomial has multiple roots (cyan points) if

p(z)=0

and

p'(z)=n

n-1

z

-1=0

. The values of

a

for roots of the second equation are called critical points (red points).

The zeros of a polynomial are continuous functions of its coefficients. If

a=0

, one root is 0, and the others are the

n-1

roots of 1. In the case of

n=5

, the roots are

0,-1,1,i-i,i

. If

a

moves in a loop from

0

to

0

, enclosing only the positive critical point (red point), the positions of the zeros corresponding to

0

and

1

of

p(z)

as a function of

a

are interchanged.