Question Number 188512 by vishal1234 last updated on 02/Mar/23
$${x}^{\mathrm{3}} +\boldsymbol{{y}}^{\mathrm{3}} +\boldsymbol{{z}}^{\mathrm{3}} −\mathrm{3}\boldsymbol{{xyz}}= \\ $$$$\left(\boldsymbol{{x}}+\boldsymbol{{y}}+\boldsymbol{{z}}\right)\:\left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} −\boldsymbol{{xy}}−\boldsymbol{{yz}}−\boldsymbol{{zx}}\right) \\ $$$$\boldsymbol{{if}}\:\boldsymbol{{x}}+\boldsymbol{{y}}+\boldsymbol{{z}}\:=\:\mathrm{0},\:{then} \\ $$$${x}^{\mathrm{3}} +\boldsymbol{{y}}^{\mathrm{3}} +\boldsymbol{{z}}^{\mathrm{3}} −\mathrm{3}\boldsymbol{{xyz}} \\ $$$$=\:\mathrm{0}×\left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} −\boldsymbol{{xy}}−\boldsymbol{{yz}}−\boldsymbol{{zx}}\right) \\ $$$$=\:\mathrm{0} \\ $$