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Question
Question 12
Verify that:
x3+y3+z3−3xyz=12(x+y+z)[(x−y)2+(y−z)2+(z−x)2]
Verify that:
x3+y3+z3−3xyz=12(x+y+z)[(x−y)2+(y−z)2+(z−x)2]
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Solution
x3+y3+z3−3xyz=12(x+y+z)[(x−y)2+(y−z)2+(z−x)2]R.H.S.=12(x+y+z)[x2+y2−2xy+y2+z2−2yz+z2+x2−2zx]=12(x+y+z)[2x2+2y2+2z2−2xy−2yz−2zx]=(x+y+z)[x2+y2+z2−xy−yz−zx]=x[x2+y2+z2−xy−yz−zx]+y(x2+y2+z2−xy−yz−zx)+z(x2+y2+z2−xy−yz−zx)=x3+xy2+xz2−x2y−xyz−zx2+yx2+y3+yz2−xy2−y2z−zxy+zx2+zy2+z3−zxy−yz2−z2xx3+y3+z3−3xyz=L.H.S.
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