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 a polynomial with a single
root of multiplicity 
is 
, which has 
as a root of multiplicity 2.
Fundamental Theorem of Algebra
See also
Degenerate, Frivolous Theorem of Arithmetic, Polynomial, Polynomial Factorization, Polynomial Roots, Principal RingExplore with Wolfram|Alpha
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.Cite this as:
Weisstein, Eric W. "Fundamental Theorem of Algebra." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FundamentalTheoremofAlgebra.html