Every polynomial equation having complex coefficients and degree  has at least one complex root. This theorem was first proven by Gauss. It is equivalent to the statement that a polynomial  of degree  has  values  (some of them possibly degenerate) for which . Such values are called polynomial roots. An example of apolynomial with a single root of multiplicity  is , which has  as a root of multiplicity 2.

REFERENCES:

Courant, R. and Robbins, H. "The Fundamental Theorem of Algebra." §2.5.4 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 101-103, 1996.

Krantz, S. G. "The Fundamental Theorem of Algebra." §1.1.7 and 3.1.4 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 7 and 32-33, 1999.

Smithies, F. "A Forgotten Paper on the Fundamental Theorem of Algebra." Notes Rec. Roy. Soc. London 54, 333-341, 2000.