Using induction for $x^n - 1$ is divisible by $x - 1$

Question

Prove using induction that for all non-negative integers n and for all integers $ x > 1 $, $ x^n - 1 $ is divisible by $ x - 1 $.

Step 1: We will prove this using induction on n. Step 2: Assume the claim is true when $ n = 1 $. $$ x^{n+1} - 1 = x(x^n - 1) + (x - 1) $$

Can someone help me with this further?

Answer 1

Hint: $$x^{n+1} - 1 = x(x^n - 1) + (x - 1)=x(x-1)(?) + (x - 1)=(x-1)(x(?) + 1)$$

Alternative: Actually We don't need to use Induction,

Let $f(x)=x^n-1$ Now $f(1)=0$, That means $1$ is a root if $f(x)$ so $f(x)=(x-1)\times(?)$ ( By '$?$' I mean some another factors as $f(x)$ is polynomial of degree $n$ )