$x = y = z = 0$ gives $P(0) = 0,$ then $z = 0$ gives $P(x) = -P(-x),$ so $P(x)$ is odd. So $P(x+y) = P(x) + P(y)$ for all reals $x,y.$ In particular, this means that $P(x) = P(1) \cdot x$ for all integers $x.$ But then $P(x) - P(1) \cdot x$ is a polynomial with infinitely many zeros, so $P(x) = P(1) \cdot x$ for all $x.$ This can be confirmed to work, and we are done.