Squaring x + y + z = 0 gives x2 + y2 + z2 + 2(xy + yz + zx) = 0. Since xy + yz + zx = 0, it simplifies to x2 + y2 + z2 = 0. Since squares are non-negative, x, y, z must be 0.
Three rational numbers x, y, z satisfy x + y + z = 0 and xy + yz + zx = 0. Show that all the rational numbers x, y, z must be simultaneously zero.
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