<s>[INST] Find $P(z)$ so that $P\left(x-\frac1x\right) = x^5 - \frac1{x^5}.$ Given that $x^n - \frac1{x^n}$ is expressible as a polynomial in $x - \frac1x$ with real coefficients only if $n$ is an odd positive integer, find $P(z)$ so that $P\left(x-\frac1x\right) = x^5 - \frac1{x^5}.$

My brain is not working with me today!  I don't know how to start this problem.  Solutions are greatly appreciated.
 [/INST] Hint:
$$(x-\frac{1}{x})^5=(x^5-\frac{1}{x^5})-5(x^3-\frac{1}{x^3})+10(x-\frac{1}{x})$$
$$(x-\frac{1}{x})^3=(x^3-\frac{1}{x^3})-3(x-\frac{1}{x})$$
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