a/(a-b)(a-c) + b/(b-c)(b-a) + c/(c-a)(c-b)

We simplify the expression involving the  three terms each being a rational algebraic expression. We also assume that the given expression is a/((a-b)(a-c)) +b/((b-c)(b-a))+c/((c-a)(c-b))

Then each ter can have an equivalent expression for each of term by taking the common dinominator as the LCM of the dinominators of the 3 terms, i.e. (a-b)(b-c)(c-a).

Then the 1st term:a/((a-b)(a-c)) = a(b-c)(-1)/{(a-b)(b-c)(c-a)}

2nd term: b/((b-c)(b-a))= b(-1)(c-a)/{(a-b)(b-c)(c-a)}

3rd term: c/((c-a)(c-b))= c(-1)(a-b)/{(a-b)(b-c)(c-a)]

Therfore, the dinominators being same we can add the numerators of the equivalent expressions.

Su of the numerators; -a(b-c)-b(c-a)-c(a-b)= -ab+ac-bc+ab-ca+bc= 0

Therefore, the sum of the three terms = 0/{(a-b)(b-c)(c-a)}=0.