Is x − 1 a factor of x100 − 1? What about x + 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 x100 − 1.
Divisor = (x − 1)
Putting x = 1 in the given polynomial:
(1)100 − 1
= 1 − 1
= 0
∴( x − 1) is a factor of the polynomial x100 − 1.
Similarly, checking for (x + 1).
To check whether (x + 1) is a factor of x100 − 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)100 − 1
= 1 − 1
= 0
∴( x + 1) is also a factor of the polynomial x100 − 1.