3. Let $x=-\sqrt{2}+\sqrt{3}+\sqrt{5}, y=\sqrt{2}-\sqrt{3}+\sqrt{5}$, and $z=\sqrt{2}+\sqrt{3}-\sqrt{5}$. What is the value of the expression below?
$$
\frac{x^{4}}{(x-y)(x-z)}+\frac{y^{4}}{(y-z)(y-x)}+\frac{z^{4}}{(z-x)(z-y)}
$$
Answer: 20
Solution: Writing the expression as a single fraction, we have
$$
\frac{x^{4} y-x y^{4}+y^{4} z-y z^{4}-z x^{4}+x z^{4}}{(x-y)(x-z)(y-z)}
$$

Note that if $x=y, x=z$, or $y=z$, then the numerator of the expression above will be 0 . Thus, $(x-y)(x-z)(y-z) \mid\left(x^{4} y-x y^{4}+y^{4} z-y z^{4}-z x^{4}+x z^{4}\right)$. Moreover, the numerator can be factored as follows.
$$
(x-y)(x-z)(y-z)\left(x^{2}+y^{2}+z^{2}+x y+y z+z x\right)
$$

Hence, we are only evaluating $x^{2}+y^{2}+z^{2}+x y+y z+z x$, which is equal to
$$
\begin{aligned}
\frac{1}{2}\left[(x+y)^{2}+(y+z)^{2}+(z+x)^{2}\right] & =\frac{1}{2}\left[(2 \sqrt{5})^{2}+(2 \sqrt{2})^{2}+(2 \sqrt{3})^{2}\right] \\
& =2(5+2+3) \\
& =20
\end{aligned}
$$