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Fundamental Theorem of Algebra

There are a couple of ways to state the Fundamental Theorem of Algebra. One way is:

A polynomial function with complex numbers for coefficients has at least one zero in the set of complex numbers .

A different version states:

An n th degree polynomial function with complex coefficients has exactly n zeros in the set of complex numbers, counting repeated zeros .

Note: The real numbers are a subset of the complex numbers, since every real number can be written in a + b i form, with b = 0 . So, the theorem is also true for polynomials with real coefficients.

Example:

g ( x ) = x 3 − 2 x 2 + 9 x − 18

In this case, the coefficients are all real numbers: 3 , − 2 and 9 .

Set g ( x ) = 0 and factor over the complex numbers to find the zeros.

0 = x 2 ( x − 2 ) + 9 ( x − 2 ) 0 = ( x − 2 ) ( x 2 + 9 ) 0 = ( x − 2 ) ( x + 3 i ) ( x − 3 i ) x = 2  or  x = − 3 i  or  x = 3 i