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Question Number 188512 by vishal1234 last updated on 02/Mar/23

x^3 +y^3 +z^3 −3xyz=  (x+y+z) (x^2 +y^2 +z^2 −xy−yz−zx)  if x+y+z = 0, then  x^3 +y^3 +z^3 −3xyz  = 0×(x^2 +y^2 +z^2 −xy−yz−zx)  = 0

$${x}^{\mathrm{3}} +\boldsymbol{{y}}^{\mathrm{3}} +\boldsymbol{{z}}^{\mathrm{3}} −\mathrm{3}\boldsymbol{{xyz}}= \\ $$$$\left(\boldsymbol{{x}}+\boldsymbol{{y}}+\boldsymbol{{z}}\right)\:\left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} −\boldsymbol{{xy}}−\boldsymbol{{yz}}−\boldsymbol{{zx}}\right) \\ $$$$\boldsymbol{{if}}\:\boldsymbol{{x}}+\boldsymbol{{y}}+\boldsymbol{{z}}\:=\:\mathrm{0},\:{then} \\ $$$${x}^{\mathrm{3}} +\boldsymbol{{y}}^{\mathrm{3}} +\boldsymbol{{z}}^{\mathrm{3}} −\mathrm{3}\boldsymbol{{xyz}} \\ $$$$=\:\mathrm{0}×\left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} −\boldsymbol{{xy}}−\boldsymbol{{yz}}−\boldsymbol{{zx}}\right) \\ $$$$=\:\mathrm{0} \\ $$

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