Given a symmetric bilinear form $f:V\times V \to K$ , where $V$ is a vectorspace and $K$ is an appropriate field, define the quadratic form $q:V \to K$ as $q(v):= f(v,v)$.
The Polarisation Formula states that $f(x,y) = 1/2\big( q(x+y) - q(x) - q(y)\big)$, which is easily proven.
This means , that any symmetric bilinear form $f:V\times V \to K$ is fully determined by the values $f(v,v)$ for all $v \in V$.
I now want to prove the following theorem: Prove that any symmetric $k$-linear form $M:V\times\cdots \times V \to K$ is determined by the values $M[v]^k := M[v,...,v]$ for all $v\in V$.
How does that work?