5. Cubic identity for three variables

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 \(a^3 + b^3 + c^3 -3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ac)\).