Expert: Paul Klarreich - 3/5/2010

Question
How do I factor out a (1-r) from (1-r^9)?
It is for a question in geometric sequences.
Chris

Answer
Questioner: Chris
Country: Canada
Category: Advanced Math
Private: Yes <<<<  I CORRECTED THIS.
Subject: Geometic Sequences Grade 12
Question: How do I factor out a (1-r) from (1-r^9)?
It is for a question in geometric sequences.
Chris
..........................................
In general, since you have learned the Factor Theorem, (standard 12th grade material) you know that if:

P(r) = 1 - r^9, or in general,  1 - r^n

and that P(1) = 1 - 1^n = 0,

then 1-r is a factor of P(r).

so you could use synthetic division:

- r^9 + 1  divided by  r - 1 (we fix later)

- 1  ..................................     1   (subst 1)
     -1  -1
------------------------------------------------
- 1   -1  -1 .........................-1   zero

So you have - (1 + r + r^2 +....+ r^8)

Now we fix, since we got  1-r backwards earlier:

1 - r^9 = (1-r)(1 + r + r^2 +....+ r^8)

and, of course, you observe that 1 + r + r^2 +....+ r^8  is a G.S.