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If x=a+b, y=a omega+b omega^2,z=a omega^...

If `x=a+b, y=a omega+b omega^2,z=a omega^2+bomega` show that `x^2+y^2+z^2=6ab`

Text Solution

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`L.H.S.=x^2+y^2+z^2`
`=(a+b)^2+(aomega+bomega^2)^2(aomega^2+bomega)^2`
`=a^2+b^2+2ab+a^2omega^2+b^2omega^4+2abomega^3+a^2omega^4+b^2omega^2+2abomega^3`
`=a^2+a^2omega^2+a^2omega+b^2+b^2omega+b^2omega^2+2ab+2ab+2ab`
`=a^2(1+omega^2+omega)+b^2(1+omega+omega^2)+6ab`
`=0+0+6ab=6ab=R.H.S."(Proved)"`
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