It is to my knowledge that polynomials may have roots in the complex plane. What I would like to know is if the mean of the real terms of all the roots (real and complex) of a polynomial function (i.e. P(x)) is equal to the mean of the real terms of all the roots (real and complex) of the derivative of that function (i.e. P'(x)).
My doubt rises due to the following post: The average of the roots of a polynomial equals the average of the roots of its derivative. I think the post is a bit obscure regarding complex roots, and the answer could bring some light to the matter.
The statement is true whether the roots are real or complex.
Note that the sum of roots of $$ax^n+bx^{n-1} +...=0 $$ is $\frac {-b}{a}$ regardless of complex or real roots. Thus the average is $$A_1=\frac {-b}{na}$$
The derivative is $$anx^{n-1} +(n-1)bx^{n-2}+...=0 $$ thus the sum of the roots are $\frac {-(n-1)b}{na}$ and since we have only $n-1$ roots the average is $$A_2=\frac {-(n-1)b}{n(n-1)a} = \frac {-b}{na}$$
So the averages are the same.