Show that for all real numbers (x/y/z) satisfying x+y+z=0 and xy +yz+zx=−3 the value of expression x^3 y+y^3 z +z^3 x is a constant


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Question Number 123824 by Snail last updated on 28/Nov/20

$S h o w t h a t f o r a l l r e a l n u m b e r s (x / y / z) s a t i s f y i n g$ $x + y + z = 0 a n d x y + y z + z x = - 3 t h e v a l u e o f$ $e x p r e s s i o n x^{3} y + y^{3} z + z^{3} x i s a c o n s t a n t$
Commented by Snail last updated on 28/Nov/20

$I t h i n k u h a v e d o n e a m i s t a k e$ $u h a v e w r i t t e n$ $x^{3} y + y^{3} z + z^{3} x = A (l e t) a n d {x y}^{3} + {y z}^{3} + x^{3} z = B (l e t)$ $t h e n u h a v e d o n e$ $A = - 18 - B \Rightarrow (a)$ $a n d t h e n B = - 18 - A$ $B u t t h a t d o e s n o t i m p l y A = B$
	Answered by MJS_new last updated on 28/Nov/20

	$z = - (x + y)$ $\Rightarrow$ $x y + y z + z x = - (x^{2} + x y + y^{2}) \land x^{3} y + y^{3} z + z^{3} y = - {(x^{2} + x y + y^{2})}^{2} = - 9$
	Commented by Dwaipayan Shikari last updated on 29/Nov/20

	$z = - (x + y)$ $x y + y z + x z = x y + y (- x - y) - (x + y) x = - (x^{2} + x y + y^{2})$ $x^{3} y + y^{3} z + z^{3} y$ $= x^{3} y - y^{3} (x + y) - {(x + y)}^{3} y$ $= - {(x^{2} + x y + y^{2})}^{2} = - 9$
	Commented by Snail last updated on 29/Nov/20

	$I {c a n}^{'} t u n d e r s t a n d t h e p r o c e s s$
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