What is \frac{x^{4}}{(x^{2}-y^{2})(x^{2}-z^{2})}+\frac{y^{4}}{(y^{2}-x^{2})(y^{2}-z^{2})}+\frac{z^{4}}{(z^{2}-x^{2})(z^{2}-y^{2})} equal to ?

Explanation

Let a=x^2, b=y^2, and c=z^2. The expression is \frac{a^2}{(a-b)(a-c)} + \frac{b^2}{(b-a)(b-c)} + \frac{c^2}{(c-a)(c-b)}. By Lagrange's interpolation formula or cyclic summation identity, this standard algebraic sum evaluates exactly to 1.

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