See the discussion [above](https://mathoverflow.net/a/382552).

Also, if $f(x) = x^n + \dotsb + a_{n-1}x + a_n$, then

- for $n = 2$, $\operatorname{Res}(f(x), f(-x)) = 2^2a_2a_1^2$

- for $n = 3$, $\operatorname{Res}(f(x), f(-x)) = 2^3a_3(a_3 - a_1a_2)^2$

- for $n = 4$, $\operatorname{Res}(f(x), f(-x)) = 2^4a_4(a_4a_1^2 - a_1a_2a_3 + a_3^2)^2$.