Problem 2

Let

P_1(x) = x^2 - 2

and

P_j(x) = P_1(P_{j-1}(x))

for

j = 2, 3, \ldots

. Prove that, for any positive integer

n

, the roots of the equation

P_n(x) = x

are all real and distinct.

Step 6 of 6: Match the count with the degree

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In plain words

A degree- $d$ polynomial equation has at most $d$ roots; finding exactly $d$ distinct real ones by direct construction is the cleanest possible way to prove that every root is real and simple, with nothing left unaccounted for.

\deg P_n = 2^n

Detailed analysis

Each application of $P_1$ squares the leading term, so $P_n$ has degree $2^n$ and $P_n(x)-x=0$ is a polynomial equation of degree $2^n$ (an induction on the degree confirms this). It has at most $2^n$ roots counted with multiplicity, and Step 5 exhibits exactly $2^n$ distinct real values of $x$ satisfying it. Hence all $2^n$ roots are accounted for, are real, and are pairwise distinct.