Results 1 to 2 of 2

Math Help - complex roots

  1. #1
    Member
    Joined
    Nov 2006
    Posts
    142

    complex roots

    I know how to show that,

    z^(n+1) = (z-1)[z^n + z^(n-1) + ... + 1).

    But how do I use this to find the complex roots of

    z^4 + z^3 + z^2 + z + 1 = 0 ?

    Thanks for any help in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by PvtBillPilgrim View Post
    I know how to show that,

    z^(n+1) - 1 = (z-1)[z^n + z^(n-1) + ... + 1). Mr F says: The correction is in red.

    But how do I use this to find the complex roots of

    z^4 + z^3 + z^2 + z + 1 = 0 ?

    Thanks for any help in advance.
    z^5 - 1 = 0 => z^5 = 1 => z = the fifth roots of 1.

    But z^5 - 1 = (z-1)(z^4 + z^3 + ... + 1).

    Therefore the fifth roots of 1 are solutions to (z-1)(z^4 + z^3 + ... + 1) = 0.

    Therefore the solutions to z^4 + z^3 + ... + 1 = 0 are the non-real fifth roots of 1.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Complex 5th Roots
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: February 6th 2011, 10:41 PM
  2. complex roots
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: May 20th 2010, 09:53 PM
  3. Complex Roots
    Posted in the Pre-Calculus Forum
    Replies: 8
    Last Post: August 18th 2009, 06:50 PM
  4. Complex roots
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 3rd 2009, 06:06 AM
  5. Complex Roots
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: March 22nd 2008, 08:16 PM

Search Tags


/mathhelpforum @mathhelpforum