The Fundamental Theorem of Algebra states that a polynomial equation of degree $n$ over the complex numbers has exactly $n$ (possibly duplicated) roots.

Actually the theorem states that a polynomial equation has at least one root. However, the factor theorem says that if $a$ is a root of $p(x)$ then $(x-a)$ is a divisor of $p(x).$ Therefore we can write $p(x)=(x-a)q(x)$ where $q(x)$ is a polynomial of lower degree, and the process can be repeated. Thus we can see that we must have exactly $n$ roots.

From wikipedia:

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with zero imaginary part. The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division.

Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.
In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the real numbers (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time when the study of algebra was mainly concerned with the solutions of polynomial equations with real or complex coefficients.

AGentleIntroductionToTimeComplexity AlgebraicGeometry AlgebraicNumber BasinOfAttraction CubeRoot CubicPolynomial DifferentiatingPolynomials DiophantineEquation FactorTheorem Function GaloisTheory HamiltonCycle IntegralCalculus MathematicsTaxonomy NewtonsMethod QuadraticPolynomial RemainderTheorem SquareRoot Surd TranscendentalNumber	(none)			(none)
↓	↓		↓	↓
Polynomial Root			RootsOfPolynomials
			↓
		⇌	You are here FundamentalTheoremOfAlgebra
			↓
			ComplexNumber Divisor FactorTheorem RealNumber
↓	↓		↓	↓
CubeRoot NewtonsMethod RiemannHypothesis RootsOfPolynomials SquareRoot	(none)			AddingComplexNumbers AlgebraicNumber ArgandDiagram CategoryMetaTopic CauchySequence ComplexConjugate ComplexPlane ContinuedFraction DedekindCut DividingComplexNumbers Euler ImaginaryNumber Integer IrrationalNumber MultiplyingComplexNumbers PolarRepresentationOfAComplexNumber RationalNumber RemainderTheorem TranscendentalNumber

Local neighbourhood - D3

Last change to this page
Full Page history
Links to this page

Edit this page
(with sufficient authority)
Change password

Recent changes
All pages
Search

Fundamental Theorem Of Algebra

Navigation

You are here

Local neighbourhood - D3