Respuesta :
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.
