$x y (x^{2} - y^{2}) + y z (y^{2} - z^{2}) + z x (z^{2} - x^{2})$ can be represented symbolically as?

Σ x y z (x^{2} - y^{2})

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Σ x y (x^{2} - y^{2})

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Σ x y (x - y)

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Σ x y (x y - z)

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Solution

The correct option is B $Σ x y (x^{2} - y^{2})$
$f (x) = x y (x^{2} - y^{2}) + y z (y^{2} - z^{2}) + z x (z^{2} - x^{2})$
Subsitute x=y
$f (y) = y y (y^{2} - y^{2}) + y z (y^{2} - z^{2}) + z y (z^{2} - y^{2})$
$f (y) = 0 + y^{3} z - z^{3} y + z^{3} y - y^{3} z = 0$
Subsitute x=z
$f (z) = z y (z^{2} - y^{2}) + y z (y^{2} - z^{2}) + z z (z^{2} - z^{2}) = y z^{3} - z y^{3} + y^{3} z - y z^{3} = 0$
So,
We can say that $\sum x y (x^{2} - y^{2}) = x y (x^{2} - y^{2}) + y z (y^{2} - z^{2}) + z x (z^{2} - x^{2})$

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