Question
$x^3 + y^3 + z^3 - 3xyz$ is:
✓
Answer
$x^3 + y^3 + z^3 - 3xyz = x^3 + y^3 + 3x^2y + 3xy^2 + z^3 - 3xyz - 3x^2y - 3xy^2$
$= (x + y)^3 +z^3 - 3xy(x + y + z)$
$= (x + y + z) ((x + y)^2 + z^2 - (x + y)z) - 3xy(x + y + z)$
$= (x + y + z) (x^2 + 2x^2y + y^2 + z^2 - xy - xz - 3xy)$
$= (x + y + z) (x^2 + y^2 + z^2 - xy - yz - zx)$
$= (x + y)^3 +z^3 - 3xy(x + y + z)$
$= (x + y + z) ((x + y)^2 + z^2 - (x + y)z) - 3xy(x + y + z)$
$= (x + y + z) (x^2 + 2x^2y + y^2 + z^2 - xy - xz - 3xy)$
$= (x + y + z) (x^2 + y^2 + z^2 - xy - yz - zx)$
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