Suppose that $P\in\mathbb{R}[x_1,\ldots,x_n]$ is a multilinear polynomial and suppose that $$ P=Q_1Q_2\cdots Q_n $$ where each $Q_i\in\mathbb{R}[x_1,\ldots,x_n]$ is non-constant.
Comment: We say that a polynomial $P(x_1,\ldots,x_n)$ is multilinear it each of its monomials is a constant times a product of distinct variables. For example, $xy+2z+xz$ is multilinear.
The problem: I want to prove that the factors $Q_i$ are
(a) unique (up to a permutation of factor and multiplication by constants)
(b) themself multilinear
(c) have disjoint variables.
My Attempt:
Regarding (a), I guess that it follows from the fact that $\mathbb{R}[x_1,\ldots,x_n]$ is UFD.
Regarding (b) and (c), suppose that $x_i$ is a variable in $P$. If we assign $x_j=1$ to all other variables, then we get an identity $$ P(x_i)=Q_1(x_i)Q_2(x_i)\cdots Q_n(x_i). $$ in $\mathbb{R}[x_i]$. Since $P$ is multilinear, it follows that the degree of $x_i$ is $1$. Therefore, $x_i$ appears exactly in one of the $Q_i$, and in degree $1$.
Is that proof holds?