Let $P(x)$ and $Q(x)$ be distinct, non-constant polynomials such that
\[P(Q(x)) = P(x) Q(x)\]for all $x.$  If $P(1) = P(-1) = 100,$ then find the polynomial $Q(x).$
Question:
Let $P(x)$ and $Q(x)$ be distinct, non-constant polynomials such that
\[P(Q(x)) = P(x) Q(x)\]for all $x.$  If $P(1) = P(-1) = 100,$ then find the polynomial $Q(x).$
Answer:
Let's think step by step.

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Let $P(x)$ and $Q(x)$ be distinct, non-constant polynomials such that
\[P(Q(x)) = P(x) Q(x)\]for all $x.$  If $P(1) = P(-1) = 100,$ then find the polynomial $Q(x).$
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