How do you divide #(x^m - 1) / (x - 1)#?
1 Answer
May 20, 2016
If
#(x^m-1)/(x-1) = x^(m-1)+x^(m-2)+...+x+1 = sum_(k=0)^(m-1) x^k#
Explanation:
Notice that if
#(x-1) sum_(k=0)^(m-1)x^k#
#= x sum_(k=0)^(m-1)x^k - sum_(k=0)^(m-1)x^k#
#= sum_(k=1)^m x^k - sum_(k=0)^(m-1) x^k#
#= x^m + color(red)(cancel(color(black)(sum_(k=1)^(m-1)x^k))) - color(red)(cancel(color(black)(sum_(k=1)^(m-1) x^k))) - 1#
#= x^m - 1#
Dividing both ends by
#sum_(k=0)^(m-1)x^k = (x^m-1)/(x-1)#