What is \frac{(a-b)^{2}}{(b-c)(c-a)}+\frac{(b-c)^{2}}{(c-a)(a-b)}+\frac{(c-a)^{2}}{(a-b)(b-c)}-3 equal to, where a\neq b\neq c?

Explanation

Let x=a-b, y=b-c, z=c-a, so x+y+z=0. The expression becomes \frac{x^3+y^3+z^3}{xyz} - 3. Since x+y+z=0, x^3+y^3+z^3=3xyz, making the term 3. Hence, 3 - 3 = 0.

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