Question

Prove that x + 1 is a factor of xⁿ − 1 for every odd number n.

Answer 1

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)

To check whether (x + 1) is a factor of xⁿ − 1, we have to convert (x + 1) in a form that is suitable for the application of the Factor theorem.

(x + 1) = {(x − (−1)}

Putting x = −1 in the given polynomial:

(−1)ⁿ − 1

= −1 − 1 ( (−1)ⁿ = −1, for every odd number n)

= −2

∴( x + 1) is a factor of the polynomial xⁿ − 1 for every odd number n.