How do you find the value of $N$ given $P(N) = N+51$ and other information about the polynomial $P(x)$? Problem:
Let $P(x)$ be a polynomial with integer coefficients such that $P(21)=17$, $P(32)=-247$, $P(37)=33$. If $P(N) = N + 51$ for some positive integer $N$, then find $N$.

I can't think of anyway to begin this question so any help will be appreciated.

$ \overbrace{{\rm Note}\,\ \ p(x)\!=\!x\!-\!4\,\ {\rm at}\ x\!=\!21,37}^{\textstyle{\rm so\ let}\ \ p(x)\!-x\!+\!4\, =:\, f(x)\qquad\!}\,$ so $\,f\,$ has roots $\,\!21,37\,$ $\Rightarrow$ $\,f^{\phantom{|^|}}\!\!\!\! = (x\!-\!21)(x\!-\!37)g,\,$ $g\in\Bbb Z[x]\,$ so evaluating at $\,x\!=\!n\Rightarrow$ $\ \color{#c00}{55}^{\phantom{|^|}}\!\!\! = f(n) = (\color{#90f}{n\!-\!21})(\color{#0a0}{n\!-\!37})g(n),\,$ so test $\rm\color{#90f}{fac}\color{#0a0}{tors}$ of $\color{#c00}{55}$ differing by $16^{\phantom{|^|}}\!$ 
