# Roots of a Polynomial with Complex Coefficients

Roots of a Polynomial with Complex Coefficients

The fundamental theorem of algebra states that a polynomial of degree with complex coefficients has values for which . The are called the roots of , with some of them possibly repeated. Thus the polynomial may be factored into linear terms as , where is some complex number. In the case of real coefficients, the roots are real or come in conjugate pairs.

P(z)

n

n

z

i

P(z)=0

z

i

P(z)

P(z)=(z-)(z-)…(z-)

a

n

z

1

z

2

z

n

a

n

This Demonstration considers polynomials of the form +++, with , , complex. The roots are shown as white dots at the centers of black patches.

m

1

z

a

2

m

2

z

a

3

m

3

z

a

4

m

4

z

a

2

a

3

a

4