Sum of non-real roots of equation? What is the sum of all non-real, complex roots of this equation -
$$x^5 = 1024$$
Also, please provide explanation about how to find sum all of non real, complex roots of any $n$ degree polynomial. Is there any way to determine number of real and non-real roots of an equation?

Please not that I'm a high school freshman (grade 9). So please provide simple explanation. Thanks in advance!
 A: By the fundamental theorem of algebra, the polynomial $a_0 + a_1x + \cdots + a_nx^n$ has $n$ (not necessarily distinct) roots, say $\alpha_1,\ldots,\alpha_n$. The polynomial $\frac{a_0}{a_n} + \frac{a_1}{a_n}x + \cdots + x^n$ also has roots $\alpha_1,\ldots,\alpha_n$. Factoring the latter polynomial and expanding,
\begin{align*}
\frac{a_0}{a_n} + \frac{a_1}{a_n}x + \cdots + \frac{a_{n-1}}{a_n} x^{n-1} +  x^n &= (x - \alpha_1)\cdots(x - \alpha_n) \\
&= x^n - (\alpha_1 + \cdots+ \alpha_n)x^{n-1} + \cdots.
\end{align*}
Comparing coefficients, we have $\alpha_1 + \cdots + \alpha_n = -\frac{a_{n-1}}{a_n}$, giving us an explicit formula for the sum of the roots.
We are looking for the sum of the non-real roots of $x^5 - 1024$. Clearly $x = 4$ is a root, and since $5$ is odd, there are no other real roots (in the complex plane, the roots lie on a circle with radius 4 centered at the origin in the shape of a pentagon with a vertex at $(4,0)$). The sum of all the roots is 0, so the sum of the non-real roots is $-4$.
A: Hint: By Vieta, sum of all roots is $0$, and the only real root is $4$.
P.S. For a simple argument that there is only one real root, apply Descartes' rule of signs on $x^5-1024$.
A: $\bf{My\; Solution::}$ Given $$x^5 = 1024 = 2^{10} = 4^5\Rightarrow x^5-4^5 = 0$$
So $$(x^5-4^5) = (x-5)\cdot (x^4+x^3\cdot 4+x^2\cdot 4^2+x\cdot 4^3+4^4) = 0$$
Above we use the formula $\displaystyle x^n-a^n = (x-a)\cdot (x^{n-1}+x^{n-2}\cdot a+x^{n-3}\cdot a^2+.......+a^{n-1})$
So We Get $x=4(\bf{Real \; Root})$ and $$x^4+x^3\cdot 4+x^2\cdot 4^2+x\cdot 4^3+4^4=0$$
So Sum of Roots in $\bf{Second}$ Equation is $\displaystyle = -\frac{4}{1} = -4$ 
Above we use the Formula $\displaystyle \bf{Sum\; of \; Roots} = -\frac{\bf{coeff.\; of \; x^3}}{\bf{coeff.\; of \; x^4}}$
