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