Example 2 Factorize $f(x, y, z)=x^{3}(y-z)+y^{3}(z-x)+z^{3}(x-y)$.

 

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Example 2 Factorize $f(x, y, z)=x^{3}(y-z)+y^{3}(z-x)+z^{3}(x-y)$.