Question:
If \[ (a+b+c)\{(a-b)^2+(b-c)^2+(c-a)^2\} = k(a^3+b^3+c^3-3abc) \] find: \[ k \]
Solution:
Using identity:
\[ (a+b+c)\{(a-b)^2+(b-c)^2+(c-a)^2\} = 2(a^3+b^3+c^3-3abc) \]
Comparing with
\[ (a+b+c)\{(a-b)^2+(b-c)^2+(c-a)^2\} = k(a^3+b^3+c^3-3abc) \]
We get:
\[ k=2 \]