MHB Show x³y+y³z+z³x is a constant

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For real numbers x, y, z satisfying x+y+z=0 and xy+yz+zx=-3, the expression x³y+y³z+z³x is proven to be a constant. The proof involves substituting z with -x-y, simplifying the expression, and demonstrating that it does not change with different values of x and y. Various algebraic manipulations show that the resulting expression remains invariant under the given conditions. The discussion emphasizes the importance of these constraints in establishing the constancy of the expression. Ultimately, the expression x³y+y³z+z³x holds a fixed value under the specified conditions.
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Show that for all real numbers $x,\,y,\,z$ such that $x+y+z=0$ and $xy+yz+zx=-3$, the expression $x^3y+y^3z+z^3x$ is a constant.
 
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anemone said:
Show that for all real numbers $x,\,y,\,z$ such that $x+y+z=0$ and $xy+yz+zx=-3$, the expression $x^3y+y^3z+z^3x$ is a constant.
We have (given)
$x+y+z=0 \cdots(1)$
$xy+yz+zx=-3\cdots(2)$
From (1)
$(x+y+z)^2 = x^2 + y^2 + z^2 + 2(xy +yz + zx) = 0$
or $x^2 + y^2 + z^2 + 2(-3) = 0$
or $x^2 + y^2 + z^2 = 6\cdots(3)$
Further from (1)
$x+y = -z\cdots(4)$
$y+z = -x\cdots(5)$
$z+x = -y\cdots(6)$

Now let us prove that
$x^3y+ y^3 z + z^3 x = xy^3 + yz^3 + zx^3\cdots(7)$

to prove the same
$x^3y+ y^3 z + z^3 x - (xy^3 + yz^3 + zx^3)$
$= (x^3y - xy^3) + (y^3z - yz^3) + (z^3 x - zx^3)$
$= xy(x^2 - y^2) + yz(y^2 - z^2) + zx(z^2 - x^2)$
$=xy(x+y)(x-y) + yz(y+z)( y-z) + zx(z+x)(z-x)$
$=xy(-z)(x-y) + yz(-x) (y-z) + xz(-y) (z-x)$ using (4), (5), (6)
$= - xyz(x-y) - xyz(y-z) - xyz(z-x)$
= 0so (7) is true

Now $(x^2 + y^2 + z^2)(xy + yz + zx) = 6 * (-3) $ putting values from above
Or $x^3y + x^2yz + zx^3 + xy^3 + y^3 z + y^2zx + z^2yx + yz^3 + z^3 x = 18$
or $(x^3y + y^3 z + z^3x) + (xy^3 + yz^3 + zx^3 ) + (x^2yz + xy^2z + xyz^2) = - 18$
or $(x^3y + y^3 z + z^3x) + (x^3y + y^3z + z^3x ) + (x^2yz + xy^2z + xyz^2) = - 18$ (from (7)
or $2(x^3y + y^3 z + z^3x) + xyz(x+y+z) = - 18$
or $2(x^3y + y^3 z + z^3x) + xyz. 0 = - 18$ from (1)
or $2(x^3y + y^3 z + z^3x)= - 18$ from (1)
or $(x^3y + y^3 z + z^3x) = - 9$

Which is a constant

Hence proved
 
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