Given that \(x^n - (1/x^n)\) is expressible as a polynomial in \(x - (1/x)\) with real coefficients only if \(n\) is an odd positive integer, find \(P(z)\) so that \(P(x-(1/x)) = x^5 - (1/x)^5.\)

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Given that \(x^n - (1/x^n)\) is expressible as a polynomial in \(x - (1/x)\) with real coefficients only if \(n\) is an odd positive integer, find \(P(z)\) so that \(P(x-(1/x)) = x^5 - (1/x)^5.\)

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For some reason I first thought is to factor x^5-(1/x)^5
hmm I don't think I like that either I will continue to think

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If that helps: \[ x^5\left(x^5-\frac{1}{x^5}\right)=x^{10}-1 \]
\[P(x-\frac{1}{x})=x^5-\frac{1}{x^5} \\ \text{ Let } z=x-\frac{1}{x} \\ zx=x^2-1 \\ x^2-zx-1=0 \\ x=\frac{ z \pm \sqrt{z^2+4}}{2} \\ P(z)=(\frac{z+\sqrt{z^2+4}}{2})^5-(\frac{2}{z+\sqrt{z^2+4}})^5\] I don't know which choice in x I should have made and P doesn't look like a polynomial but maybe it can be fix up I guess might be worth looking at what @thomas5267 said
\[ P(x)=x^5+ax^4+bx^3+cx^2+dx+e\\ \left(x-\frac{1}{x}\right)^5=x^5-5x^3+10x-10x^{-1}+5x^{-3}-x^{-5}\\ \left(x-\frac{1}{x}\right)^4=x^4-4x^2+6-4x^{-2}+x^{-4}\\ \left(x-\frac{1}{x}\right)^3=x^3-3x+3x^{-1}-x^{-3}\\ \left(x-\frac{1}{x}\right)^2=x^2-2+x^{-2}\\ x-\frac{1}{x}\\ \left(x-\frac{1}{x}\right)^5+a\left(x-\frac{1}{x}\right)^4+b\left(x-\frac{1}{x}\right)^3\\+c\left(x-\frac{1}{x}\right)^2+d\left(x-\frac{1}{x}\right)+e\\ =x^5+x^{-5}\\ a=0\\ b=5\\ c=0\\ d=5\\ e=0 \] Inelegant, but it works.

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