prove that x cube + y cube + z cube = ( x + y + z ) ( x square + y square + z square -xy - yz - zx )
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x^3+y^3+z^3 is the LHS
RHS
=(x+y+z) (x^2+y^2+z^2-xy-yz-zx)
=x(x^2+y^2+z^2-xy-yz-zx) + y(x^2+y^2+z^2-xy-yz-zx) + z(x^2+y^2+z^2-xy-yz-zx)
=x^3+xy^2+xz^2-x^2y-xyz-x^2z + x^2y+y^3+yz^2-xy^2-y^2z-xyz + x^2z+y^2z+z^3-xyz-yz^2-xz^2
after solving we get
x^3+y^3+z^3
LHS=RHS
HOPE IT IS HELPFUL
RHS
=(x+y+z) (x^2+y^2+z^2-xy-yz-zx)
=x(x^2+y^2+z^2-xy-yz-zx) + y(x^2+y^2+z^2-xy-yz-zx) + z(x^2+y^2+z^2-xy-yz-zx)
=x^3+xy^2+xz^2-x^2y-xyz-x^2z + x^2y+y^3+yz^2-xy^2-y^2z-xyz + x^2z+y^2z+z^3-xyz-yz^2-xz^2
after solving we get
x^3+y^3+z^3
LHS=RHS
HOPE IT IS HELPFUL
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