Prove the following for integers

Question

How can I show that
$a_{i_r}^n-a_{i_r}^{n-1}\displaystyle\sum_{i=1}^na_{i_1}+a_{i_r}^{n-2}\displaystyle\sum_{i_1<i_2}^na_{i_1}a_{i_2}-\cdots-a_{i_r}\displaystyle\sum_{i_1<i_2<\dots<i_{n-1}}^na_{i_1}a_{i_2}\cdots a_{i_{n-1}}+a_{i_1}a_{i_2}\cdots a_{i_{n-1}}a_{i_n}=0,$
$r=1,2,\dots,n$,
where $a_i\in \mathbb Z$.

Answer 1

Hint (if I have read the question right): consider the polynomial $p(x)=(x-a_1)(x-a_2)\dots (x-a_r)$

[not sure from the question what to put in the brackets - perhaps substitute $a_{i_r}$ for $a_r$]