I am an engineering student and I can't solve this problem. The drawback I have is how to work with the exponent 10. I know it is little I did but I have come this far:
Problem:
Let be $z_1,z_2$ solutions of the equation $x^2+bx+c=0$ and $b,c \in \mathbb R$. Prove that $z_1^{10}+z_2^{10} \in \mathbb C_R$
Demonstration:
If the roots are real, then $z_1^{10}+z_2^{10} \in \mathbb C_R$. End.
If one of the roots is complex, then the other is also complex and is its conjugate, by the Complex conjugate root theorem. (There cannot be a real root and a complex one). By Vieta's Formula, I have:
$z_1+z_2=-b$
$z_1z_2=c$
From here, I don't know how to continue. Does anyone know how to solve it?
Best Answer
There is no need to compute $z_1, z_2$ or their powers explicitly. All you need is that the complex conjugate of a product is equal to the product of the complex conjugates (the complex conjugate is “distributive” over addition, subtraction, multiplication and division).
As you figured out, $z_1$ and $z_2$ are either both real numbers, or complex conjugates of each other, and you already solved it for the first case.
In the second case is $$ \overline{z_1^{10}} = \overline{z_1}^{10} = z_2^{10} $$ so that $z_1^{10}$ and $z_2^{10}$ are also complex conjugates, and their sum is a real number.