What are the factors of $x (y^{2} - z^{2}) + y (z^{2} - x^{2}) + z (x^{2} - y^{2})$ ?

(y - x) (z - y) (x - z)

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(y + x) (z - y) (x + z)

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(y - x) (z + y) (x + z)

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(y - z) (z - y) (x + z)

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Solution

The correct option is C $(y - x) (z - y) (x - z)$
If we use $y = z$
then expression becomes 0, hence $(y - z)$ is a factor
Ifwe use $z = y$
then expression becomes 0, hence $(z - y)$ is a factor
If we use $x = z$
then expression becomes 0, hence $(x - z)$ is a factor
Hence $(y - x) (z - y) (x - z)$ is a factor

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∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{1}{z} & \frac{1}{z} & - \frac{(x + y)}{z^{2}} - \frac{(y + z)}{x^{2}} & \frac{1}{x} & \frac{1}{x} - \frac{y (y + z)}{x^{2} z} & \frac{x + 2 y + z}{x z} & - \frac{y (x + y)}{x z^{2}} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

\frac{x (y + z - x)}{log x} = \frac{y (z + x - y)}{log y} = \frac{z (z + x - y)}{log z},

x^{y} y^{x} = z^{y} y^{z} = x^{z} z^{x}

x^{2} + y^{2} + x y = 9, z^{2} + x^{2} + x z = 4, y^{2} + z^{2} + y x = 1.

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