Rational polynomial

Algebra Level 4

\(f(x)\) and \(g(x) \) are monic polynomials with integer coefficients such that

\[\frac{f(x)}{g(x)} =\frac{x^3+8}{x^5-2x^4+4x^3} \]

and \( \gcd(f(x), g(x) ) = 1 \). What is the value of \( f(4) + g(4) \)?

Details and assumptions

You may use the fact that \( \frac{ f(4) } { g(4) } = \frac{3}{32} \).

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