<s>[INST] Develop $(\frac{2}{x}-x)^8$ $(\frac{2}{x}-x)^8$
How do I develop this?
My book gives me four options to what that one of the terms of the expression is supposed to look like after the expression is developed. The right option is:$$^8C_3*(\frac{2}{x})^3*(-x)^5$$
I even tried graphing both expressions on my graphical calculator and the graphs are different. Could this be a mistake from my book? If not, how do I solve this?
 [/INST] The function is:
$$ f(x) = \left( \frac{2}{x} - x \right)^8 = \frac{(2-x^2)^8}{x^8} = x^8 \, \left(\frac{2}{x^2} - 1 \right)^8. $$
These have the expansions:
\begin{align}
f(x) &= \sum_{k=0}^{8} \binom{8}{k} \, \left(- \frac{2}{x}\right)^{8-k} \, (-x)^k \\
&= \frac{1}{x^8} \, \sum_{k=0}^{8} \binom{8}{k} \, 2^{8-k} \, (-1)^k \, x^{2k}
\end{align}
other expansions follow in a similar manor.
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