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27*x*^{3} − *y*^{3} − *z*^{3} − 9*xyz*

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#### Solution

The given expression to be factorized is

`27x^3 - y ^3 z^3 - 9xyz`

This can be written in the form

`27x^3 - y^3 - z^3 - 9xyz = (3x)^3 + (-y)^3+(-z)^3 -3 (3x)(-y)(-z)`

Recall the formula

`a^3+b^3 +c^3 -3abc = (a+b+c)(a^2 +b^2 +c^2 - ab - bc -ca)`

Using the above formula, we have

`27x^3 - y^3 -z^3 - 9xyz`

`={3x +(-y) + (-z)}{(3x)^2 + (-y)^2+ (-z)^2 - (3x)(-y)-(-y)(-z )- (-z)(3x)}`

` = (3x-y -z)(9x^2 +y^2 +z^2 + 3xy - yz + 3zx)`

We cannot further factorize the expression.

So, the required factorization is of `27x^3 - y^3-z^3 -9xyz` is `(3x -y-z)(9x^2 +y^2 +z^2 +3xy -yz +3zx)`.

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