Prove that x-1 is a factor of x^n-1 for any positive integer n.

Accepted Solution

Answer:    [tex]x-1[/tex] is a factor of [tex]x^n - 1[/tex] Step-by-step explanation:[tex]x-1[/tex] is a factor of [tex]x^n - 1[/tex]We will prove this with the help of principal of mathematical induction.For n = 1, [tex]x-1[/tex] is a factor [tex]x-1[/tex], which is true.Let the given statement be true for n = k that is [tex]x-1[/tex] is a factor of [tex]x^k - 1[/tex].Thus, [tex]x^k - 1[/tex] can be written equal to  [tex]y(x-1)[/tex], where y is an integer.Now, we will prove that the given statement is true for n = k+1[tex]x^{k+1} - 1\\=(x-1)x^k + x^k - 1\\=(x-1)x^k + y(x-1)\\(x-1)(x^k + y)[/tex]Thus, [tex]x^k - 1[/tex] is divisible by [tex]x-1[/tex].Hence, by principle of mathematical induction, the given statement is true for all natural numbers,n.