Question

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

Answer 1

Answer:    

$x-1$ is a factor of $x^n - 1$

Step-by-step explanation:

$x-1$ is a factor of $x^n - 1$

We will prove this with the help of principal of mathematical induction.

For n = 1, $x-1$ is a factor $x-1$, which is true.

Let the given statement be true for n = k that is $x-1$ is a factor of $x^k - 1$.

Thus, $x^k - 1$ can be written equal to  $y(x-1)$, where y is an integer.

Now, we will prove that the given statement is true for n = k+1

$x^{k+1} - 1\\=(x-1)x^k + x^k - 1\\=(x-1)x^k + y(x-1)\\(x-1)(x^k + y)$

Thus, $x^k - 1$ is divisible by $x-1$.

Hence, by principle of mathematical induction, the given statement is true for all natural numbers,n.