Theory:

Identity \(VIII\): a3+b3+c33abc=a+b+ca2+b2+c2abbcac
 
\(21x^3 + 8y^3 + z^3 - 12xyz\)
 
Let us write the expression as \(21x^3 + 8y^3 + z^3 - 12xyz\).
 
Using the identity \(VIII\): a3+b3+c33abc=a+b+ca2+b2+c2abbcac
 
21x3+8y3+z312xyz=3x+2y+z9x2+4y2+z2(3x)(2y)(2y)(z)(3x)(z)
 
21x3+8y3+z312xyz=3x+2y+z3x2+2y2+z26xy2yz3xz
 
Important!
If a3+b3+c3=0
 
Then, a3+b3+c33abc=a+b+ca2+b2+c2abbcac
 
We know that, a3+b3+c3=0
 
a3+b3+c33abc=0a2+b2+c2abbcac
 
a3+b3+c33abc=0a3+b3+c3=3abc