Question:
\[ (a-b)^3+(b-c)^3+(c-a)^3= \]
(a) \[ (a+b+c)(a^2+b^2+c^2-ab-bc-ca) \]
(b) \[ (a-b)(b-c)(c-a) \]
(c) \[ 3(a-b)(b-c)(c-a) \]
(d) none of these
Solution:
Let
\[ x=a-b \]
\[ y=b-c \]
\[ z=c-a \]
Then
\[ x+y+z=0 \]
Using identity:
\[ x^3+y^3+z^3=3xyz \quad \text{when} \quad x+y+z=0 \]
Therefore,
\[ (a-b)^3+(b-c)^3+(c-a)^3 \]
\[ =3(a-b)(b-c)(c-a) \]
Hence, the correct answer is:
\[ \boxed{3(a-b)(b-c)(c-a)} \]