Checking whether B cubed has zero drift

Let $B$ be a standard Brownian motion. Is the process $B_t^3$ a martingale? Justify your answer.

No. By Ito, $d(B_t^3)=3B_t^2\,dB_t+3B_t\,dt$, which has a non-zero drift. Concretely $\mathbb{E}[B_t^3\mid\mathcal F_s]=B_s^3+3(t-s)B_s\ne B_s^3$. The corrected process $B_t^3-3tB_t$ is a martingale.