NL-NWO 1990 Problem 3

A polynomial $f(x)=ax^4+bx^3+cx^2+dx$ with $a,b,c,d>0$ is such that $f(x)$ is an integer for $x \in \{ -2,-1,0,1,2 \}$ and $f(1)=1$ and $f(5)=70$ . $(a)$ Show that $a=\frac{1}{24}, b=\frac{1}{4},c=\frac{11}{24},d=\frac{1}{4}$ . $(b)$ Prove that $f(x)$ is an integer for all $x \in \mathbb{Z}$ .