Question

Is x − 1 a factor of x¹⁰¹ − 1? What about x + 1?

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 x¹⁰¹ − 1

Divisor = (x − 1)

Putting x = 1 in the given polynomial:

(1)¹⁰¹ − 1

= 0

∴( x − 1) is a factor of the polynomial x¹⁰¹ − 1.

Similarly, checking for x + 1.

To check whether (x + 1) is a factor of x¹⁰¹ − 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)¹⁰¹ − 1

= −1 − 1

= −2

∴( x + 1) is not a factor of the polynomial x¹⁰¹ − 1.