Results 1 to 2 of 2

Math Help - Reducibility of a polynomial

  1. #1
    Member
    Joined
    Jan 2010
    Posts
    104

    Reducibility of a polynomial

    Let f(x)=x^5-120x^4+1152x^3+341x^2-1185x-21945

    The roots of this, as presented by the computer, happen to be

    3.041032416, 109.44694126, 10.55305874, -1.520516208 +/- 1.983912452 i

    Using only the roots, is f(x) reducible in Z[x]?

    I'm a little confused how to go about this...do you just multiply together possible combinations and see if integers result, or is there a more efficient way?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by twittytwitter View Post
    Let f(x)=x^5-120x^4+1152x^3+341x^2-1185x-21945

    The roots of this, as presented by the computer, happen to be

    3.041032416, 109.44694126, 10.55305874, -1.520516208 +/- 1.983912452 i

    Using only the roots, is f(x) reducible in Z[x]?

    I'm a little confused how to go about this...do you just multiply together possible combinations and see if integers result, or is there a more efficient way?
    Call those roots a,\, b,\, c,\, d\pm ei respectively. Clearly none of the linear factors x-a,\, x-b,\, x-c,\,x-d-ei,\,x-d+ei is in Z[x]. So if there is a factorisation in Z[x] it must consist of a quadratic factor and a cubic factor. The two complex linear factors must obviously belong in the same real factor, hence so must their product (x-d-ei)(x-d+ei) = x^2 -2dx+ d^2+e^2. But 2d is not an integer, so that quadratic expression is not in Z[x]. Notice however that a=-2d, which raises the possibility that the cubic factor (x-a)(x^2 -2dx+ d^2+e^2) might be in Z[x]. If so, then the quadratic factor would be the product (x-b)(x-c) = x^2 - (b+c)x + bc of the remaining two linear factors. Use your calculator to find b+c and bc, to see whether they look like integers.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. reducibility in Q implies reducibility in Z.
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 5th 2011, 05:46 AM
  2. True or False? Regarding reducibility...
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: May 11th 2010, 10:32 AM
  3. Replies: 1
    Last Post: December 15th 2009, 07:26 AM
  4. [SOLVED] dividing polynomial by a polynomial
    Posted in the Algebra Forum
    Replies: 1
    Last Post: February 3rd 2008, 02:00 PM
  5. dividing a polynomial by a polynomial
    Posted in the Algebra Forum
    Replies: 1
    Last Post: August 2nd 2005, 12:26 AM

Search Tags


/mathhelpforum @mathhelpforum