I'm looking at the trilinear example in this Wikipedia article. I would like to translate this example to the case $f:R\times R\times R\rightarrow R$, where $R$ is the real number line. The basis vector set for each vector space $R^1=R$ is $\{\textbf{e}\}=\{(1)\}$. The article seems to claim that for any $\textbf{a}=(a),\textbf{b}=(b),\textbf{c}=(c)\in R^1$, $f(\textbf{a},\textbf{b},\textbf{c})=f(a,b,c)$ can be expressed in the form $abc\times f(\textbf{e},\textbf{e},\textbf{e})=abc\times f(1,1,1)$.
Suppose the multilinear function is $f(x,y,z)=xy+z$. Then $f(2,3,4)=10$. Also, $f(\textbf{e},\textbf{e},\textbf{e})=f(1,1,1)=2$. So according to the article, the answer to $f(2,3,4)$ should be $2\times 3\times 4\times f(1,1,1)=24\times 2 = 48$. But $48$ is incorrect because $48\neq 10$. So I must have incorrectly translated the formula. What is the correct translation?
$f$ is a multilinear polynomial, but not a multilinear map. While both must form a straight line in each variable when all others are held constant, a multilinear map must additional obey the linearity homomorphism.