If a homogeneous multivariate polynomial is zero at all points, is it necessarily the zero polynomial?

Question

A very stupid question:

If a homogeneous polynomial $P(x_1,\cdots,x_k)=0$ for all $x \in \mathbb{R}^k$ then does that mean it is the zero polynomial? (I have proven this using induction but I suspect this is something rather trivial and perhaps comes from fundamental theorem).