Results 1 to 3 of 3

Math Help - Polynomial of fifth degree

  1. #1
    Newbie
    Joined
    Jul 2006
    Posts
    12

    Polynomial of fifth degree

    Seems easy, but how would you show that every polynomial of fifth degree has at least one real root?

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

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,749
    Thanks
    650
    Hello, OntarioStud!

    How would you show that every polynomial of fifth degree has at least one real root?

    Graphically, maybe?


    Given: . f(x) \:=\:ax^5 + bx^4 + cx^3 + dx^2 + ex + f

    For positive a: \;\lim_{x\to\infty}f(x) \,=\,\infty and \lim_{x\to-\infty}f(x) \,=\,-\infty *


    Since the polynomial function is continuous, it must cross the x-axis somewhere.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    * .Of course, the argument hold for negative a as well.

    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    In complex analysis it is shown that any non-constant polynomial is reducible to linear ploynomials over the complex field. Furthermore, the real polynomials are reducible to linear factors such that they come in conjugate pairs of complex numbers. Since there is an odd number of factors it is not possible to a polynomial into non real complex numbers. Thus the factor (x-r) appears where r is real. This assures us that there is at least one real solution.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Polynomial of Degree 2
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: January 19th 2011, 02:20 PM
  2. polynomial of degree 3
    Posted in the Algebra Forum
    Replies: 2
    Last Post: March 5th 2010, 06:28 AM
  3. low degree polynomial
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: December 14th 2009, 01:40 AM
  4. 7th degree polynomial
    Posted in the Algebra Forum
    Replies: 1
    Last Post: January 24th 2009, 07:05 AM
  5. 4th degree polynomial
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 27th 2005, 08:41 PM

Search Tags


/mathhelpforum @mathhelpforum