Results 1 to 2 of 2

Math Help - Show polynomials in Zn[x] as products of linear factorizations

  1. #1
    Newbie
    Joined
    May 2013
    From
    United States
    Posts
    3

    Show polynomials in Zn[x] as products of linear factorizations

    I have two questions on this topic. The first one is finding the x3+2 as a product of linear factors in Z3[X]. I have in my notes that -

    x3+2=x3-1 (in mod 3)
    =(x)3-(1)3
    =(x3-1)(x2+x+1)
    =(x-1)(x2+x+1)
    =(x-1)(x-1)(x-1)

    The part of this I dont get is the second and third step. How do we go from x3-1 to (x)3-(1)3=(x3-1)(x2+x+1)?

    The second question is to show 2x3+3x-7x-5 as a product of linear factors in Z1[x]. I keep running into polynomials that cant be factored. I presume the step Im missing is just like the previous question where i need to use the properties of mod, but Im not sure how/where to do it?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    GJA
    GJA is offline
    Member
    Joined
    Jul 2012
    From
    USA
    Posts
    109
    Thanks
    29

    Re: Show polynomials in Zn[x] as products of linear factorizations

    Hi jquinc21,

    The reason you're having trouble seeing the second step is because it's not true. If you multiply out (x^{3} -1)(x^{2}+x+1) you will have a polynomial of degree 5 in \mathbb{Z}_{3}[x], which is not equal to the degree 3 polynomial x^{3}-1.

    Does this clear things up? Let me know if there is a point of confusion.

    Also, not sure if you have seen this or not, but it's worth checking out the "Freshmen's Dream" theorem Freshman's dream - Wikipedia, the free encyclopedia. Since 3 is a prime this theorem tells us that IN \mathbb{Z}_{3}[x], (x-1)^{3}=x^{3}-1, which was what you concluded in your post above.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: January 27th 2013, 02:24 PM
  2. Show fnfinite products are equal
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 17th 2010, 11:06 AM
  3. factorizations
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: December 2nd 2009, 02:56 AM
  4. factorizations
    Posted in the Math Software Forum
    Replies: 0
    Last Post: December 1st 2009, 05:23 PM
  5. dot products and linear functionals
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: July 4th 2008, 10:42 AM

Search Tags


/mathhelpforum @mathhelpforum