Q:

write abc-ABC in terms of linear combination of products

It is easy to see the following identity: $ab-AB=(a-A)b+A(b-B)$. What about $abc-ABC$?
Can we write $abc-ABC$ as a linear combination of special terms? Here a special term means a product of three parts and at least two parts in the product are of the form $a-A$, $b-B$, $c-C$ and the possible remaining part is the remaining letter. Examples of special terms are $(a-A)(b-B)C$, $A(b-B)(c-C)$,$(a-A)b(c-C)$, $(a-A)(b-B)(c-C)$, $(a-A)(b-B)c$.
Note that there are only finitely many special terms so in theory we can set up an equation and solve for the coefficients. But this requires a lot of work. I doubt the existent of such a linear combination. Do you have a clever way to solve the problem? 

A:

We suppose $A,B,C$ are distinct non-zero elements in an integral domain.
Consider $P(a,b,c) = abc - ABC$ as a polynomial with variables a$,b,c$. Suppose it can be written as a linear combination of special terms. Note that if we set $a=A$ and $b=B$, all special terms vanish. However we get $P(A,B,c) =AB(c-C)$. Simply choosing $c \neq C$, we get a contradiction. So there is no such special linear combination that expresses $P(a,b,c)$.

