conjugated roots of equation

The rules


concerning the complex conjugatesMathworldPlanetmath of the sum and product of two complex numbersMathworldPlanetmathPlanetmath, may be by induction generalised for arbitrary number of complex numbers wk. Since the complex conjugate of a real number is the same real number, we may write


for real numbers ak(k=0, 1, 2,). Thus, for a polynomialPlanetmathPlanetmathP(x):=a0xn+a1xn-1++an  we obtain


I.e., the values of a polynomial with real coefficients computed at a complex number and its complex conjugate are complex conjugates of each other.

If especially the value of a polynomial with real coefficients vanishes at some complex number z, it vanishes also at z¯.  So the roots of an algebraic equation


with real coefficients are pairwise complex conjugate numbers.

Example. The roots of the binomial equation


are  x=1,  x=-1±i32,  the third roots of unityMathworldPlanetmath.

Title conjugated roots of equation
Canonical name ConjugatedRootsOfEquation
Date of creation 2013-03-22 17:36:51
Last modified on 2013-03-22 17:36:51
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Topic
Classification msc 12D10
Classification msc 30-00
Classification msc 12D99
Synonym roots of algebraic equation with real coefficients
Related topic PartialFractionsOfExpressions
Related topic QuadraticFormula
Related topic ExampleOfSolvingACubicEquation