A respect for polynomials

Algebra Level 5

Let \(f(x)\) and \(g(x)\) be monic cubic polynomials, such that

  • \(f(x)+g(x) = p (x)\),

  • \(p(1) = 13\),

  • \(p(3) = 97\),

  • \( p(5) = 349\),

  • \(f(7) = 491\),

  • \(g(19) = 6938 \), and

  • \(g(x) \) is a depressed cubic polynomial.

Evaluate \(f(30) + g (27)\).

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