(x-y)^3-(z-y)^3+(z-x)^3

Question

(x-y)^3-(z-y)^3+(z-x)^3

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Answer 1

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