No polynomial with integer coefficients satisfies the conditions

Question

Prove that there is no polynomial with integer coefficients $P(x)$ such that $P(7)=5$ and $P(15)=9$.

In general; how to know if there exists a polynomial with integer coefficients $P(a)=b$ and $P(c)=d$, where $a,b,c$, and $d$ are all integers?

Answer 1

For each $a,b\in \mathbb{Z}$ and $P\in \mathbb{Z}[X]$ we have $$a-b\mid P(a)-P(b)$$ so $$15-7\mid P(15)-P(7)= 9-5\implies 8\mid 4$$

A contradiction!

On the other hand if $$a-c\mid b-d$$ then such a polynomial exist. Take $$P(x) ={d-b\over c-a} (x-a)+b$$