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Question

Show that for all real numbers x, y, z such that x+y+z=0 and xy+yz+zx=3, the expression x3y+y3z+z3x is a constant

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Solution

Consider the equation whose roots are x, y, z :
(tx)(ty)(tz)=0
This gives t33t=λ0, where λ=xyz. Since x, y, z are roots of this equation, we have
x33xλ=0,y33yλ=0,z33zλ=0
Multiplying the first by y, the second by z and the third by x, we obtain
x33xyλy=0
y33yzλz=0
z33zxλx=0
Adding we obtain
x3y+y3z+z3x+z3x3(xy+yz+zx)λ(x+y+z)=0
This simplifies to
x3y+y3z+z3x=9
(Here one may also solve for y and z in terms of x and substitute these values in x3y+y3z+z3x to get -9).

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