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If-x-a-2-bc-y-b-2-ca-z-c-2-ab-then-prove-that-x-3-y-3-z-3-3xyz-is-a-perfect-square-

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Question Number 179157 by Agnibhoo98 last updated on 25/Oct/22
If x = a^2 − bc, y = b^2 − ca, z = c^2 − ab then prove that, x^3 + y^3 + z^3 − 3xyz is a perfect square.
Ifx=a2−bc,y=b2−ca,z=c2−abthenprovethat,x3+y3+z3−3xyzisaperfectsquare.
Answered by som(math1967) last updated on 25/Oct/22
x−y=a^2 −bc−b^2 +ca =(a−b)(a+b) +c(a−b) =(a−b)(a+b+c) y−z=(b−c)(a+b+c) z−x=(c−a)(a+b+c) x^3 +y^3 +z^3 −3xyz =(x+y+z)(x^2 +y^2 +z^2 −xy−yz−zx) =(1/2)(x+y+z){(x−y)^2 +(y−z)^2 +(z−x)^2 } =(a^2 +b^2 +c^2 −ab−bc−ca) ×[(a+b+c)^2 {(a−b)^2 +(b−c)^2 +(c−a)^2 } =(a+b+c)(a^2 +b^2 +c^2 −ab−bc−ca) ×(1/2)(a+b+c){(a−b)^2 +(b−c)^2 +(c−a)^2 } =(a^3 +b^3 +c^3 −3abc)(a^3 +b^3 +c^3 −3abc) =(a^3 +b^3 +c^3 −3abc)^2 perfect square
x−y=a2−bc−b2+ca=(a−b)(a+b)+c(a−b)=(a−b)(a+b+c)y−z=(b−c)(a+b+c)z−x=(c−a)(a+b+c)x3+y3+z3−3xyz=(x+y+z)(x2+y2+z2−xy−yz−zx)=12(x+y+z){(x−y)2+(y−z)2+(z−x)2}=(a2+b2+c2−ab−bc−ca)×[(a+b+c)2{(a−b)2+(b−c)2+(c−a)2}=(a+b+c)(a2+b2+c2−ab−bc−ca)×12(a+b+c){(a−b)2+(b−c)2+(c−a)2}=(a3+b3+c3−3abc)(a3+b3+c3−3abc)=(a3+b3+c3−3abc)2perfectsquare

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