The rules $$\overline{w_1+w_2} = \overline{w_1}+\overline{w_2} \quad\mbox{and}\quad \overline{w_1w_2} = \overline{w_1}\,\overline{w_2},$$ 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 $$\overline{a_kz^k} = a_k\overline{z}^k$$ for real numbers $a_k\,\, (k = 0,\,1,\,2,\,\ldots)$ . Thus, for a polynomial $P(x) := a_0x^n+a_1x^{n-1}+\ldots+a_n$ we obtain $$\overline{P(z)} = \overline{a_0z^n+a_1z^{n-1}+\ldots+a_n} = {a_0\overline{z}^n+a_1\overline{z}^{n-1}+\ldots+a_n} = P(\overline{z}).$$ 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.