conjugated roots of equation

The rules

\overline{w_{1}+w_{2}}=\overline{w_{1}}+\overline{w_{2}}\quad\mbox{and}\quad% \overline{w_{1}w_{2}}=\overline{w_{1}}\,\overline{w_{2}},

concerning the complex conjugates of the sum and product of two complex numbers, may be by induction generalised for arbitrary number of complex numbers $w_{k}$ . Since the complex conjugate of a real number is the same real number, we may write

\overline{a_{k}z^{k}}=a_{k}\overline{z}^{k}

for real numbers $a_{k}\,\,(k=0,\,1,\,2,\,\ldots)$ . Thus, for a polynomial $P(x):=a_{0}x^{n}+a_{1}x^{n-1}+\ldots+a_{n}$ we obtain

\overline{P(z)}=\overline{a_{0}z^{n}+a_{1}z^{n-1}+\ldots+a_{n}}={a_{0}% \overline{z}^{n}+a_{1}\overline{z}^{n-1}+\ldots+a_{n}}=P(\overline{z}).

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 $\overline{z}$ . So the roots of an algebraic equation

P(x)=0

with real coefficients are pairwise complex conjugate numbers.

Example. The roots of the binomial equation

x^{3}\!-\!1=0

are $x=1$ , $x=\frac{-1\pm{i}\sqrt{3}}{2}$ , the third roots of unity.

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