Can someone help me with this math question?

Question

write a paragraph in which you dicuss wheter x 1 is always a factor of P(x)=x^n-1, where n is a positive interger.

thanks so much!

Answer 1

If x-1 is a factor then x=1 is a root. P(1) = 1^n -1 =0.

Since 1 is a root of P(x), then x-1 is a factor.

Also, when you divide P(x) by x-1 you get the following quotient without remainder:

x^ (n-1) + x^(n-2) + ... + x + 1