WOLFRAM|DEMONSTRATIONS PROJECT

The Fundamental Theorem of Algebra

​
variables
integer
real
a
7
0
0
a
6
0
0
a
5
0
0
a
4
0
0
a
3
1
0
a
2
-
6
0
a
1
4
0
a
0
-
2
0

3
x
-6
2
x
+4x-2
The fundamental theorem of algebra: a polynomial of degree
n
with complex coefficients has
n
complex roots, counting multiplicity. Euler, Gauss, Lagrange, Laplace, Weierstrass, Smale, and many other mathematicians worked to prove this theorem, using complex analytic, topological, or algebraic methods.
Let
p(z)=
a
7
7
z
+
a
6
6
z
+
a
5
5
z
+
a
4
4
z
+
a
3
3
z
+
a
2
2
z
+
a
1
z+
a
0
. This Demonstration lets you vary the coefficients of
p(z)
. The roots (or zeros) of
p
are shown as red points in a contour plot of the absolute value of
p(z)
in the complex plane. Contour lines of the absolute values
1
8
,
1
4
,
1
2
,1,2,4,…
of
p(z)
circle these roots. Mouseover a root to display its approximate value. Click a green-boxed zero to set that variable to zero.