Question 71
A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities
$$
P(x, y, z) = P(x, y, x y - z) = P(x, z x - y, z) = P(y z - x, y, z)
$$
Prove that there exists a polynomial $F(t)$ in one variable such that
$$
P(x, y, z) = F\left(x^{2} + y^{2} + z^{2} - x y z\right)
$$
$$
P(x, y, z) = P(x, y, x y - z) = P(x, z x - y, z) = P(y z - x, y, z)
$$
Prove that there exists a polynomial $F(t)$ in one variable such that
$$
P(x, y, z) = F\left(x^{2} + y^{2} + z^{2} - x y z\right)
$$