conjugated roots of equation

The rules

$$\overline{w_1+w_2}=\overline{w_1}+\overline{w_2}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}\overline{w_1{w}_2}=\overline{w_1}\overline{w_2},$$

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 $a_k(k=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots })$. Thus, for a polynomial  $P(x):=a_0{x}^n+a_1{x}^{n-1}+\mathrm{\dots }+a_n$  we obtain

$$\overline{P(z)}=\overline{a_0{z}^n+a_1{z}^{n-1}+\mathrm{\dots }+a_n}=a_0{\overline{z}}^n+a_1{\overline{z}}^{n-1}+\mathrm{\dots }+a_n=P(\overline{z}).$$

I.e., the values of a polynomial with real coefficients computed at a complex number and its complex conjugate arecomplex 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