Results 1 to 6 of 6

Math Help - Reducible

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    21

    Reducible

    Let F be a field and let f(x) exist in F[x] such that deg f(x) = 3. Prove that f(x) is reducible in F[x] if and only if f(x) has a root in F.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    It would help if you stated what you were having trouble with.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2010
    Posts
    21
    Well I can visualize this proof with graphs, but I am unable write it in a proof. It makes sense that something that is reducible with a deg of 3 would cross the x-axis 2 times, which would make it reducible. But all of this is not formal. How do I begin?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    You can try one direction at a time. Suppose that f is reducible. This means that you can write f(x)=g(x)h(x) where the degree of g and h is at least 1 and less than 3. Can you say something about the degrees of g and h in particular?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Apr 2010
    Posts
    21
    well since deg f(x) is 3 the deg g(x) and deg h(x) can only be either 1 or 3, since exponents are added together when multiplying. So the forward method is making more sense but going the other direction with the roots throws me off. I know that f(x) will cross the axis 3 times but how does that declare that it is reducible?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    The degrees actually have to be 1 and 2. Since you have a degree 1, it is of the form x-a for some a. Therefore, a root is x=a.

    For the other direction, you should recall the factor theorem. If f(x) is a polynomial and f(a)=0 (i.e. a is a root), then f(x)=(x-a)g(x) for some polynomial g. The result is immediate.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. reducible polynomial x^p+a in Zp
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 27th 2010, 11:53 PM
  2. Reducible Polynomials
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: October 14th 2010, 12:12 AM
  3. [SOLVED] Reducible Equations
    Posted in the Pre-Calculus Forum
    Replies: 6
    Last Post: July 14th 2010, 03:56 AM
  4. Reducible problem. Help!
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: April 18th 2010, 02:33 AM
  5. R[x] - Reducible?
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: October 18th 2009, 08:27 PM

Search Tags


/mathhelpforum @mathhelpforum