Question 83
Suppose $p$ is the polynomial
$$
p(z) = az^4 + bz^3 + cz^2 + dz + e,
$$
where $a$, $b$, $c$, $d$, $e$ are complex numbers with $a \neq 0$. Prove that there is a complex number $h$ such that $p(h+z) = p(h-z)$ for all complex numbers $z$, if and only if
$$
b^3 - 4abc + 8a^2d = 0.
$$
$$
p(z) = az^4 + bz^3 + cz^2 + dz + e,
$$
where $a$, $b$, $c$, $d$, $e$ are complex numbers with $a \neq 0$. Prove that there is a complex number $h$ such that $p(h+z) = p(h-z)$ for all complex numbers $z$, if and only if
$$
b^3 - 4abc + 8a^2d = 0.
$$