Results 1 to 2 of 2
Like Tree1Thanks
  • 1 Post By emakarov

Thread: prove f is a rational polynomial

  1. #1
    Newbie
    Joined
    Oct 2013
    From
    German
    Posts
    1

    prove f is a rational polynomial

    $\displaystyle f(x)$ is polynomial with complex coefficients. $\displaystyle \forall n\in Z$, $\displaystyle f(n)$is integer, prove: coefficients of $\displaystyle f(x)$ are rational numbers, and give some examples about rational case.

    ### Prove:

    ---
    * consider coefficients are integers, of course $\displaystyle f(n)$ are integers.

    * consider coefficients are rationals, we have $\displaystyle f(x)=\frac{1}{2}x(x+1)$, two consecutive integer can be divided by $\displaystyle 2$, there must be one even number.

    How about the cases?:


    * real coeffs

    * complex coeffs

    ---
    And can you give me some more examples?
    Last edited by integer; Oct 7th 2013 at 01:20 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,577
    Thanks
    790

    Re: prove f is a rational polynomial

    The following ideas are based on this article (PDF). It turns out that every integer-valued polynomial p(x) has the form

    $\displaystyle p(x) = a_0+a_1\binom{x}{1}+ \dots+ a_n\binom{x}{n}$ (*)

    Indeed, the coefficients (omitting the constant term, which is zero) of polynomials $\displaystyle \binom{x}{1}$, $\displaystyle \binom{x}{2}$, ..., $\displaystyle \binom{x}{n}$ form a triangular matrix, so these polynomials form a basis, i.e., every polynomial of degree ≤ n with no constant term can be expressed as a linear combination of these polynomials, perhaps with complex coefficients. So, every polynomials from ℂ[x] of degree ≤ n can be written in the form (*) with complex $\displaystyle a_k$. Next prove by induction on k that $\displaystyle a_k\in\mathbb{Z}$.
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: Oct 4th 2012, 10:15 PM
  2. Prove rational raised to a rational is rational.
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: Feb 15th 2011, 08:12 PM
  3. Polynomial and rational functions
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: Nov 29th 2009, 06:15 PM
  4. Replies: 6
    Last Post: May 5th 2009, 06:49 AM
  5. Polynomial and Rational Functions
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: Mar 18th 2006, 09:03 AM

Search Tags


/mathhelpforum @mathhelpforum