Cubic identity for three variables — lesson. Mathematics CBSE, Class 9.

Identity \(VIII\):

a^{3} + b^{3} + c^{3} - 3 abc = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - bc - ac)

\(21x^3 + 8y^3 + z^3 - 12xyz\)

Let us write the expression as \(21x^3 + 8y^3 + z^3 - 12xyz\).

Using the identity \(VIII\):

a^{3} + b^{3} + c^{3} - 3 abc = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - bc - ac)

{21x}^{3} + {8y}^{3} + z^{3} - 12 xyz = (3 x + 2 y + z) ({9x}^{2} + {4y}^{2} + z^{2} - (3x) (2y) - (2y) (z) - (3 x) (z))

{21x}^{3} + {8y}^{3} + z^{3} - 12 xyz = (3 x + 2 y + z) ({3x}^{2} + {2y}^{2} + z^{2} - 6 xy - 2 yz - 3 xz)

Important!

a^{3} + b^{3} + c^{3} = 0

Then,

a^{3} + b^{3} + c^{3} - 3 abc = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - bc - ac)

We know that,

a^{3} + b^{3} + c^{3} = 0

a^{3} + b^{3} + c^{3} - 3 abc = (0) (a^{2} + b^{2} + c^{2} - ab - bc - ac)

\begin{array}{l} a^{3} + b^{3} + c^{3} - 3 abc = (0) \\ a^{3} + b^{3} + c^{3} = 3 abc \end{array}

Did you find an error?

6. Cubic identity for three variables