Freaky Functions

Algebra Level 5

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

Details and assumptions

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

A non-zero polynomial is a polynomial that is not the zero polynomial. Equivalently, there is a value α\alpha such that f(α)0 f(\alpha) \neq 0 .
A non-zero polynomial is allowed to have specific values that evaluate to 0. E.g. f(x)=x1 f(x) = x-1 is a non-zero polynomial even though f(1)=0f(1) = 0 .


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