Theory:

Identity:
a3+b3+c33abc=a+b+ca2+b2+c2abbcac
Example:
Expand \(27x^3 + 8y^3 + z^3 - 18xyz\).
 
Solution:
 
Let us write the expression of \(27x^3 + 8y^3 + z^3 - 18xyz\) using the identity a3+b3+c33abc=a+b+ca2+b2+c2abbcac.
 
27x3+8y3+z312xyz=3x3+2y3+z333x2yz
 
\(=\) 3x+2y+z9x2+4y2+z2(3x)(2y)(2y)(z)(3x)(z)
 
\(=\) 3x+2y+z3x2+2y2+z26xy2yz3xz
 
Important!
If a3+b3+c3=0 then the identity a3+b3+c33abc=a+b+ca2+b2+c2abbcac is rewritten as follows:
 
a3+b3+c33abc=0a2+b2+c2abbcac
 
a3+b3+c33abc=0
 
a3+b3+c3=3abc
Example:
Evaluate \(8^3 - 5^3 - 3^3\).
 
Solution:
 
By the identity if a3+b3+c3=0, then a3+b3+c3=3abc.
 
On comparing \(8^3 - 5^3 - 3^3\) with a3+b3+c3=0 we have \(a\) \(=\) \(8\), \(b\) \(=\) \(5\) and \(c\) \(=\) \(3\).
 
Here \(a+b+c\) \(=\) \(8-5-3\) \(=\) 0
 
Therefore, \(8^3 - 5^3 - 3^3\) \(=\) \(3 \times 8 \times 5 \times 3\) \(=\) \(360\)