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Fundamental Theorem of Algebra


Every polynomial equation having complex coefficients and degree >=1 has at least one complex root. This theorem was first proven by Gauss. It is equivalent to the statement that a polynomial P(z) of degree n has n values z_i (some of them possibly degenerate) for which P(z_i)=0. Such values are called polynomial roots. An example of a polynomial with a single root of multiplicity >1 is z^2-2z+1=(z-1)(z-1), which has z=1 as a root of multiplicity 2.


See also

Degenerate, Frivolous Theorem of Arithmetic, Polynomial, Polynomial Factorization, Polynomial Roots, Principal Ring

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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

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