A respect for polynomials

Algebra Level 5

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

  • f(x)+g(x)=p(x)f(x)+g(x) = p (x),

  • p(1)=13p(1) = 13,

  • p(3)=97p(3) = 97,

  • p(5)=349 p(5) = 349,

  • f(7)=491f(7) = 491,

  • g(19)=6938g(19) = 6938 , and

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

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

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