$u(t,B_t)$ is a martingale if $u(t,x)$ is polynomial in each variables and satisfies the heat equation

I want to show that for $$u(t,x)$$ which is a polynomial in $$t$$ and $$x$$ such that $$\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial^2 u}{\partial x^2}$$ we have $$u(t,B_t)$$ is a martingale where $$B_t$$ stands for the standard Brownian motion.

Durrett's "Probability theory and examples" shows that $$E_x u(t, B_t)$$ is constant in $$t$$ and concludes right away that $$u(t,B_t)$$ is a martingale. How is this possible?

Any help is appreciated.