Look at the following polynomials and their factorizations:

Question

x^2-1=(x-1)(x+1)

x^3-1=(x-1)(x^2+x+1)

x^4-1=(x-1)(x^3+x^2+x+1)

In general, how can x^n-1 be factored. Show that this factorization works by multiplying the factors

Answer 1

It should be obvious from the examples that

x^n - 1 =
(x-1)[x^(n-1) + x^(n-2) .. x + 1]

The series in the second factor stops when the last term => x^0 = 1