I am reading an overview on Galois theory. It is mentioned that solutions are symmetric, for example, what I understood that if one solution/zero of $f(x)$ is $(a+1)$ ($a$ is a cconstant) then another solution would be $(-a-1)$? Why do we assume solutions of polynomial are symmetric? If one solution is $(a+1)$ then why could't $(2a-1)$ be a another solution? If $(2a-1)$ a solution, then does it mean $(-2a-1)$ has to be a solution also ? why?
Why does it means to be that solutions of polynomial are symmetric (please provide example if I am wrong in above)? Why don't we consider the possibility asymmetric solution? Please refer to the theorem, no need to provide the proof here.