WOLFRAM|DEMONSTRATIONS PROJECT

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.