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Fundamental theorem of algebra
1. The Fundamental Theorem of Algebra
If p(x) is any polynomial of degree n>1 with complex coefficients, then p(x)
has at least one complex zero.
A related theorem:
A polynomial of degree n has at most n zeros.
10
8
6
4
2
-2
-4
-6
-5 5 10 15
10
8
6
4
2
-2
-4
-15 -10 -5 5 10 15
f x( ) = x-2( )2
10
8
6
4
2
-2
-15 -10 -5 5
f x( ) = x-2( )2+3
3. Multiplicity of Zeros:
If a polynomial has the same zero more than once, we describe that zero’s
multiplicity:
Ex:
Another theorem:
A polynomial of degree n>1 with complex coefficients has exactly n
complex zeros, if multiplicities are counted.
xxxxp 2510)( 23
++=