HMMT 2021 A4

Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c)=$ $P(b, c, a)$ and $P(a, a, b)=0$ for all real $a, b$, and $c$. If $P(1,2,3)=1$, compute $P(2,4,8)$. Note: $P(x, y, z)$ is a homogeneous degree 4 polynomial if it satisfies $P(k a, k b, k c)=k^{4} P(a, b, c)$ for all real $k, a, b, c$.