x^3+y^3+z^3-3 x y z is: - Vidyadip

MCQ

$x^3+y^3+z^3-3 x y z$ is:

A
$(x + y + z)^3$
B
$(x - y + z)^3$
C
$(x + y + z)^3 - 3xyz$
✓
$(x+y+z)\left(x^2+y^2+z^2-x y-y z-z x\right)$

✓

Answer

Correct option: D.

$(x+y+z)\left(x^2+y^2+z^2-x y-y z-z x\right)$

$x^3+y^3+z^3-3 x y z=x^3+y^3+3 x^2 y+3 x y^2+z^3-3 x y z-3 x^2 y-3 x y^2$
$=(x+y)^3+z^3-3 x y(x+y+z)$
$=(x+y+z)\left((x+y)^2+z^2-(x+y) z\right)-3 x y(x+y+z)$
$=(x+y+z)\left(x^2+2 x^2 y+y^2+z^2-x y-x z-3 x y\right)$
$=(x+y+z)\left(x^2+y^2+z^2-x y-y z-z x\right)$

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