# Cyclic Polynomials - Factoring

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

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 $(x-y)(y-z)(z-x).\ _\square$.

**Cite as:**Cyclic Polynomials - Factoring.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/cyclic-polynomials-factoring/