A respect for polynomials

$f(x)$ and $g(x)$ being monic cubic polynomials, $g(x)$ depressed:

$f(x)=x^3+ax^2+cx+d$ ,

$g(x)=x^3+ex+f$ ,

$p(x)=f(x)+g(x)=2x^3+ax^2+(c+e)x+d+f$ .

Plugging in the given values...

$p(1)=13 \Rightarrow 2+a+c+d+e+f=13 \Rightarrow a+c+d+e+f=11$

$p(3)=97 \Rightarrow 54+9a+3c+d+3e+f=97 \Rightarrow 9a+3c+d+3e+f=43$

$\ldots$

...results in a linear system of equations, whose solution is $a=3, c=0, d=1, e=4, f=3$ , so $f(x)=x^3+3x^2+1$ and $g(x)=x^3+4x+3$ , and $f(30)+g(27)=\boxed{49495}$ .