Let \(n\) be a fixed positive integer. Find all functions \( f:\mathbb{R}\to\mathbb{R}\) satisfying \[ f(x^{n+1}-y^{n+1})=(x-y)[f(x)^n+f(x)^{n-1}f(y)+\cdots+f(x)f(y)^{n-1}+f(y)^n].\]