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Aratati ca (x-y)^3+(y-z)^3+(z-x)^3=3(x-y)(y-z)(z-x)

Răspuns :

(x-y)^3+ (y-z)^3+ (z-x)^3=x^3 –3(x^2)*y + 3xy^2 - y^3 + y^3 –3(y^2)*z + 3yz^2 - z^3 + z^3 –3(z^2)*x + 3zx^2 - x^3=3*(-(x^2)*y + xy^2– (y^2)*z + yz^2–(z^2)*x + zx^2)
-(x^2)*y + xy^2– (y^2)*z + yz^2–(z^2)*x + zx^2=-xy(x-y) +z(x^2 – y^2) –(z^2)(x-y)=(x-y)(-xy+zx+zy-z^2)= (x-y)((-x)(y-z)+z(y-z))=(x-y)(y-z)(z-x)

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