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(Roots of unity ) x n -1=(x-1) p (x) where p (x) is a polynomial of degree n-1 where p (x) is a 0 +a 1 x+...+a n-1 x n-1 . I need to solve for the coefficients a i . My attempt at solving for coefficients: (x n -1)=a 0 x+a 1 x 2 +...+a n-1 x n -a 0- a 1 x-...-a n-1 x n-1 where I get a 0 =1 and a n-1 =1 I do not know if this is right. I don't think I solved for all the coefficients. Please help me.

(Roots of unity )
xn-1=(x-1)p(x) where p(x) is a polynomial of degree n-1
where p(x) is a0+a1x+...+an-1xn-1. I need to solve for the coefficients ai. 
My attempt at solving for coefficients:
(xn-1)=a0x+a1x2+...+an-1xn-a0-a1x-...-an-1xn-1
where I get a0=1 and an-1=1 
I do not know if this is right. I don't think I solved for all the coefficients. Please help me. 

Grade:12th pass

1 Answers

Aditya Gupta
2081 Points
5 years ago
one easy way is tot simply divide x^n – 1 by x – 1 using the ususal division algorithm and then find out the coeffs. however, from the sum of GP formula, we know that 
1+x+x^2+x^3+....+x^(n–1)= 1*(x^n – 1)/(x – 1)
or xn-1=(x-1)(1+x+x^2+x^3+....+x^(n–1))
hence we conclude that all the coefficients are equal to 1.

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