-3
$\begingroup$

How can I use partial-fraction decomposition for this fraction?

$f(x)=\dfrac{1}{1+x^{7}}$

1 Answers 1

1

The roots of $x^7+1$ are $\{-1, -\zeta, -\zeta^2,-\zeta^3,-\zeta^4,-\zeta^5,-\zeta^6 \}$ where $\zeta = \cos(2\pi/7)+i\sin(2\pi/7)$.

  • 0
    how you arrive in this solution? Can you give me your method? Thanks – 2017-01-15
  • 0
    These are the classic roots of unity as expressed as roots of the [cyclotomic polynomials](https://en.wikipedia.org/wiki/Cyclotomic_polynomialhttps://en.wikipedia.org/wiki/Cyclotomic_polynomial). – 2017-01-15
  • 0
    [This](https://en.wikipedia.org/wiki/Root_of_unity) is an easier reference. – 2017-01-15
  • 0
    I don't know what you are saying to me. Can you do in a sheet of paper and send to me, how you get all the roots? – 2017-01-15
  • 0
    Did you click on the links? – 2017-01-15
  • 0
    Yes, but I don't know how to aply the theory to this exercise. – 2017-01-15
  • 0
    Use the fact that $\zeta^k = \cos(2\pi\cdot k/7)+i\sin(2\pi\cdot k/7) $ for $k = 0, \ldots,6$. You can leave the cos and sin expressions or calculate them with an ordinary calculator, depends what your prof wants. – 2017-01-15