number theory - Devise a test for divisibility of an integer by 11, in terms of properties of its digits

Using the fact that $10 \equiv-1 \pmod{11}$, devise a test for divisibility of an integer by $11$, in terms of properties of its digits.

Approach:

Let the number with its digits $a_0\cdots a_n$ be represented as $f(x)=a_0x^n+\cdots+a_n$. By exhaustion $f(-1)=0$ if the number is divisible by $11$, so $f(10)$ is divisible by $11$. Do I have to prove my conjecture? or it's trivial.