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> Given a polynomial series $\{p\_n(x)\}\_{n=1}^{\infty}$ in $\mathbb{R}[X]$ with initial value $p\_1(x)=x^2-2$. And $p\_k(x)=p\_1(p\_{k-1}(x))=p\_{k-1}(x)^2-2,\;k=2,3,\cdots$.
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> Prove that for each integer $n$, all roots of $p\_n(x)-x$ are real and distinct.
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This is an problem in my linear algebra textbook.
I tried to figure out the relation of the roots between adjacent polynomial, but I couldn't find any useful result. Also I thought if it could be solved by induction, but it seems impracticable.