Show that for all real numbers x, y, z such that $x + y + z = 0$ and $x y + y z + z x = 3$ , the expression $x^{3} y + y^{3} z + z^{3} x$ is a constant

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Solution

Consider the equation whose roots are x, y, z :
$(t - x) (t - y) (t - z) = 0$
This gives $t^{3} - 3 t = λ 0$ , where $λ = x y z$ . Since x, y, z are roots of this equation, we have
$x^{3} - 3 x - λ = 0, y^{3} - 3 y - λ = 0, z^{3} - 3 z - λ = 0$
Multiplying the first by y, the second by z and the third by x, we obtain
$x^{3} - 3 x y - λ y = 0$
$y^{3} - 3 y z - λ z = 0$
$z^{3} - 3 z x - λ x = 0$
Adding we obtain
$x^{3} y + y^{3} z + z^{3} x + z^{3} x - 3 (x y + y z + z x) - λ (x + y + z) = 0$
This simplifies to
$x^{3} y + y^{3} z + z^{3} x = - 9$
(Here one may also solve for y and z in terms of x and substitute these values in $x^{3} y + y^{3} z + z^{3} x$ to get -9).

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