<s>[INST] Polynomial Factoring $(2^x)+(2^{-x})=a$.  What is $(8^x)+(8^{-x})$?
I did:
$$\begin{align} (2^x)^3+(2^{-x})^3
 &= [(2^x)+(2^{-x})] [(2^x)^2-(2^x)(2^{-x})+(2^{-x})^2]
\\&= a [(2^x)^2+2(2^x)(2^{-x})+(2^{-x})^2-3(2^x)(2^{-x})]
\\&= a {[(2^x)+(2^{-x})]^2-3(2^x)(2^{-x})}
\\&= a [a^2-3(2^x)(2^{-x})]
\\&= a^3-3a(2^x)(2^{-x}) \end{align}$$
The answer is $(a^3-3a)$. But I don't know how to solve now. Please, could you help me?
 [/INST] The posted proof looks good. Alternatively, using the identity $(u+v)^3=u^3+v^3+3uv(u+v)$:
$$a^3=(2^x+2^{-x})^3=\left(2^x\right)^3+\left(2^{-x}\right)^3+3\cdot 2^x\cdot 2^{-x}(2^x + 2^{-x})=8^x+8^{-x}+ 3 \cdot 1 \cdot a$$
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