Identity of x^3 y^3 z^3 - 3xyz

To find the identity of the expression x^3 y^3 z^3 - 3xyz, we can factor it using the method of grouping. The given expression can be rewritten as:

x^3 y^3 z^3 - 3xyz = (x^3 y^3 z^3) - (3xyz)

Factor out common terms

We can factor out the common term xyz from both parts of the expression:

(x^3 y^3 z^3) - (3xyz) = xyz * (x^2 y^2 z^2 - 3)

Factor the remaining expression

Now, we need to factor the expression inside the parentheses, which is x^2 y^2 z^2 - 3. This is a difference of squares, so we can factor it as follows:

x^2 y^2 z^2 - 3 = (xy)(xy)(z^2) - 3

Apply the formula for a^2 - b^2

Using the formula for a^2 - b^2, we can rewrite the expression as:

(xy)^2 (z^2) - 3 = (xy - √3)(xy + √3)(z^2)

Combine the factors

Now, let's combine the factors we obtained. The expression becomes:

xyz * (xy - √3)(xy + √3)(z^2)

Final simplified identity

Finally, we can rewrite the expression with the common factor xyz factored out:

(x^3 y^3 z^3) - (3xyz) = (xyz)(xy - √3)(xy + √3)(z^2)

Therefore, the identity of x^3 y^3 z^3 - 3xyz is (xyz)(xy - √3)(xy + √3)(z^2).

Yes it is correct this identity is is ok
x^3 + y^3 + z^3 - 3xyz = ( x+ y+z)(x^2+y^2+z^2-xy-yz-zx)