Question 102
Find one of the polynomials $f(x, y, z)$ whose degree is $3$, with real coefficients, that satisfy the following conditions.
• $f(x, y, z) + x$ is divisible by $y + z$
• $f(x, y, z) + y$ is divisible by $z + x$
• $f(x, y, z) + z$ is divisible by $x + y$
A polynomial $P(x, y, z)$ is divisible by a polynomial $Q(x, y, z)$ means that there exists a polynomial $R(x, y, z)$ that satisfies $P(x, y, z) = Q(x, y, z)R(x, y, z)$.
• $f(x, y, z) + x$ is divisible by $y + z$
• $f(x, y, z) + y$ is divisible by $z + x$
• $f(x, y, z) + z$ is divisible by $x + y$
A polynomial $P(x, y, z)$ is divisible by a polynomial $Q(x, y, z)$ means that there exists a polynomial $R(x, y, z)$ that satisfies $P(x, y, z) = Q(x, y, z)R(x, y, z)$.