Let's do that by induction :
For \(n=1\), \(x^1-y^1\) is obviously divisible by \(x-y\)
If we assume the property holds at rank \(n\), then \(x^{n+1}-y^{n+1}=x(x^n-y^n)+y^n(x-y)\). Since \(x^n-y^n\) is divisible by \((x-y)\), we have \(A\) such that \(x^n-y^n=A(x-y)\) hence \(x^{n+1}-y^{n+1}=(x-y)(Ax+y^n)\).
Hence by induction for all \(n\ge1\), \(x-y\) divides \(x^n-y^n\)
