Prove that x − 1 is a factor of xn − 1 for every natural number n.
Factor theorem says that for the polynomial p(x) and for the number a, if we have p(a) = 0, then (x − a) is a factor of p(x).
Given: Polynomial xn − 1
Divisor = (x − 1)
Putting x = 1 in the given polynomial:
(1)n − 1
= 1 − 1 { (1)n = 1, for all natural numbers n}
= 0
∴( x − 1) is a factor of the polynomial xn − 1 for every natural number n.