Factoring: Is (x + 1) a factor of x^100 - 1? Of x^1000 - 1?

[doglover]

I don't know how to get this one started, and I have the feeling that I'm just missing something obvious.

"Is (x+1) a factor of x^100 - 1? Explain. Is (x - 1) a factor of x^1000 - 1? Explain."

If someone could help me with either I might be able to get the other on my own. Thanx!

 

[stapel]

"Is (x+1) a factor of x^100 - 1? Explain. Is (x - 1) a factor of x^1000 - 1? Explain."

Is x + 1 a factor of x[sup:25aos26m]2[/sup:25aos26m] - 1? How do you know? What result do you get when you divide x[sup:25aos26m]2[/sup:25aos26m] - 1 by x + 1?

Is x + 1 a factor of x[sup:25aos26m]3[/sup:25aos26m] - 1? How do you know? What result do you get when you divide x[sup:25aos26m]3[/sup:25aos26m] - 1 by x + 1?

Is x + 1 a factor of x[sup:25aos26m]4[/sup:25aos26m] - 1? How do you know? What result do you get when you divide x[sup:25aos26m]4[/sup:25aos26m] - 1 by x + 1?

Is x + 1 a factor of x[sup:25aos26m]5[/sup:25aos26m] - 1? How do you know? What result do you get when you divide x[sup:25aos26m]5[/sup:25aos26m] - 1 by x + 1?

Do you see any pattern? (Hint: Even versus odd; negative versus positive.)

Can you think of any way to relate "x + 1 is (or is not) a factor" to "x = -1 is (or is not) a root"? :wink:

Eliz.

 

[doglover]

Is this how I post a response? Well, the explanation was a great help. I solved both the long way, but as an extended question:
Does x^[sup:qw5qspcq](even)[/sup:qw5qspcq] - 1 divided by (x + 1) create a difference of "squares"? [ (x + 5)(x - 5) = (x[sup:qw5qspcq]2[/sup:qw5qspcq] - 25) ]