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11.Factorize \(27x^3 + y^3 + z^3 - 9xyz\)
Answer :

\(27x^3 + y^3 + z^3 - 9xyz = (3x)^3 + (y)^3 + (z)^3 - 3(3x)(y)(z)\)
\(= (3x + y +z)((3x)^2 + y^2 + z^2 - 3xy - yz - z(3x)\)
i.e., by using identity \(a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)\)
We get,
\(= (3x + y +z)(9x^2 + y^2 + z^2 - 3xy - yz - 3zx)\)
Therefore, \(27x^3 + y^3 + z^3 - 9xyz = (3x + y +z)(9x^2 + y^2 + z^2 - 3xy - yz - 3zx)\)