Question

Verify that xcube+ycube+zcube-3xyz=1/2[x+y+z](x-y)square+(y-z)square+(z-x)square

Answer 1

Answer:

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P^3+Q^3+R^3-3.P.Q.R

=(P+Q+R).(P^2+Q^2+R^2-P.Q-Q.R-R.P). ,

=(P+Q+R).[P.(P-Q)+Q.(Q-R)+R.(R-P)]. , putting the values of P, Q and R.

=(y+z-x+z+x-y+x+y-z).[(y+z-x).(y+z-x-z-x+y)+(z+x-y).(z+x-y-x-y+z)+(x+y-z).(x+y-z--y

-z+x)].

=(x+y+z).[(y+z-x).2.(y-x) +(z+x-y).2.(z-y)+(x+y-z).2.(x-z)].

= 2.(x+y+z)[(y-x)^2+z.(y-x)+(z-y)^2+x.(z-y)+(x-z)^2+y.(x-z)].

= 2.(x+y+z).[y^2-2xy+x^2+yz-zx+z^2-2yz+y^2+zx-xy+x^2-2zx+z^2+xy-yz].

= 2.(x+y+z).[2.x^2+2.y^2+2.z^2–2xy-2yz-2zx].

=2.2(x+y+z).(x^2+y^2+z^2-xy-yz-zx)