Theory:

Identity VIII: \(a^3 + b^3 + c^3 -3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ac)\)
 
Let us verify the above identity by direct multiplication of the right hand side expression.
 
\((a + b + c)(a^2 + b^2 + c^2 - ab - bc - ac)\) \(=\) \(a (a^2 + b^2 + c^2 - ab - bc - ac) + b(a^2 + b^2 + c^2 - ab - bc - ac) + c(a^2 + b^2 + c^2 - ab - bc - ac)\)
 
\(=\) \((a^3 + ab^2 + ac^2 - a^2b - abc - a^2c) + (a^2b + b^3 + bc^2 - ab^2 - b^2c - abc) + (a^2c + b^2c + c^3 - abc - bc^2 - ac^2)\)
  
Simplify the above expression by combining the like terms.
 
\(=\) \(a^3 + b^3 + c^3 - 3abc\)
 
Thus we have the identity \(a^3 + b^3 + c^3 -3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ac)\).
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+z(9x2+4y2+z2(3x)(2y)(2y)(z)(3x)(z))
 
\(=\) 3x+2y+z3x2+2y26xy2yz3xz
Important!
If \(a+b+c = 0\) 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 a+b+c=0, then a3+b3+c3=3abc.
 
On comparing \(8^3 - 5^3 - 3^3\) with a+b+c=0 we have \(a\) \(=\) \(8\), \(b\) \(=\) \(-5\) and \(c\) \(=\) \(-3\).
 
Here \(a + b + c\) \(=\) \(8 - 5 - 3\) \(=\) \(0\)
 
Therefore, \(8^3 - 5^3 - 3^3\) \(=\) \(3 \times 8 \times -5 \times -3\)
 
\(=\) \(360\)