Cyclic Polynomials - Factoring

Factorize \( x^2 (y-z) + y^2 (z-x) + z^2 (x-y) \).

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Let the above polynomial equal \( f(x, y, z) \). We know that \( f(x, y, z) \) is a cyclic polynomial because \( f(x, y, z) = f(y, z, x) \).

Note that \( f(y, y, z ) = 0 \). By the factor theorem, we can say that \( (x-y) \) is a factor of \( f(x, y, z) \). If \( (x-y) \) is a factor of \( f(x, y, z) \), then it must be true that \( (x-y)(y-z)(z-x) \) is also a factor of \( f(x, y, z) \) because \( f(x, y, z) \) is cyclic.

The degree of \( f(x, y, z) \) is 3. Also, \( (x-y)(y-z)(z-x) \) has a degree of 3. Thus, the factored result of \( f(x, y, z) \) is equal to \( m(x-y)(y-z)(z-x) \). If we let \( (x, y, z) = (0, 1, 2) \) and solve the equation \( f(0, 1, 2) = m(0-1)(1-2)(2-0) \), we can see that \( m = 1 \) and our final factored result is \( (x-y)(y-z)(z-x).\ _\square\).