Freaky Functions

Algebra Level 5

Suppose \(f(x)\) and \(g(x)\) are non-zero polynomials with real coefficients, such that \(f(g(x))=f(x)\cdot g(x).\) If \(g(2)=37,\) what is \(g(3)?\)

Details and assumptions

The zero polynomial is a polynomial that is identically 0, i.e. \(f=0\).

A non-zero polynomial is a polynomial that is not the zero polynomial. Equivalently, there is a value \(\alpha\) such that \( f(\alpha) \neq 0 \).
A non-zero polynomial is allowed to have specific values that evaluate to 0. E.g. \( f(x) = x-1\) is a non-zero polynomial even though \(f(1) = 0 \).


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