$p(x)$ be a fifth degree polynomial with integer coeffients that has an integral root $\alpha$. If $p(2)=13$ and $p(10)=5$

Question

$p(x)$ be a fifth degree polynomial with integer coeffients that has an integral root $\alpha$. If $p(2)=13$ and $p(10)=5$ then find the value of $\alpha$

I am looking for various other approaches to this problem

Thanks !

Answer 1

We have that $a-b$ divides $p(a)-p(b)$ for all integers $a$ and $b$.

$\alpha -2$ divides $0-13$ and so $\alpha -2 \in \{-13,-1,1,13 \}$.

$\alpha -10$ divides $0-5$ and so $\alpha -10 \in \{-5,-1,1,5 \}$.

Therefore, $\alpha \in \{ -11,1,3,15 \} \cap \{ 5,9,11,15 \} = \{ 15 \}$.

Answer 2

The polynomial will have a factor $(x-\alpha)$ , and it will be an integer for integral values of $x$ , now 13 and 5 can be factored as $\pm13\times \pm1$ and $\pm5\times \pm1$ , after checking various possible combinations. Only value of $\alpha$ which satisfies both the given conditions is 15

So , $\alpha=15$