I want to show that the aplication
$$p: \mathbb{R} \times \cdots \times \mathbb{R} \to \mathbb{R}$$
defined by
$$p(x_{1}, \dots, x_{n})=\prod_{i=1}^{n}x_{i}$$
is a $n$-linear form. Let's see in particular, for example, if it is bilinear
$$p(\alpha u+\beta v, w)=((\alpha u+\beta v) \cdot w)=\alpha(u \cdot w)+\beta(v \cdot w)=\alpha p(u, w)+\beta p(v, w)$$
Analogously
$$p(u, \alpha v + \beta w)=\alpha p(u, v) + \beta p(u, w)$$
For the first $n$-values, the result is even trivial, but how can I generalize this expression to run through the $n$-terms? any help is welcome!