Roots of a Polynomial with Complex Coefficients

m

1

2

3

4

5

6

7

m

2

1

2

3

4

m

3

1

2

3

m

4

0

1

a 2
a 3
a 4

axes

plot range

0.5

1

2

3

4

plot points factor

1

2

3

4

5

z

-0.1

2

z

+1

The fundamental theorem of algebra states that a polynomial

P(z)

of degree

n

with complex coefficients has

n

values

z

i

for which

P(z)=0

. The

z

i

are called the roots of

P(z)

, with some of them possibly repeated. Thus the polynomial may be factored into linear terms as

P(z)=

a

n

(z-

z

1

)(z-

z

2

)…(z-

z

n

)

, where

a

n

is some complex number. In the case of real coefficients, the roots are real or come in conjugate pairs.

This Demonstration considers polynomials of the form

m

1

z

+

a

2

m

2

z

+

a

3

m

3

z

+

a

4

m

4

z

, with

a

2

,

a

3

,

a

4

complex. The roots are shown as white dots at the centers of black patches.