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