Results 1 to 4 of 4

Thread: Proving Divisibility

  1. #1
    Newbie
    Joined
    Aug 2012
    From
    NY
    Posts
    24
    Thanks
    1

    Proving Divisibility

    The polynomial f(X) has integer coefficients and an integer root. Prove that, for every non negative integer n, the product f(0)f(1)...f(n) is divisible by (n+1)! .
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Jul 2010
    From
    Vancouver
    Posts
    432
    Thanks
    17

    Re: Proving Divisibility

    This seems like an interesting question. What have you tried so far?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2009
    Posts
    170
    Thanks
    8

    Re: Proving Divisibility

    Just some ideas ...

    The integer root is very important. You can represent a polynomial as $\displaystyle f(x) = \sum_{i=0}^n a_i x^i$ where the $\displaystyle a_i$ are integers (as stated in the problem). How would you represent the polynomial if you knew one of its roots, say $\displaystyle r$?


    What would happen to the product $\displaystyle f(0)f(1) ... f(n)$ if $\displaystyle r \geq 0$ (i.e. $\displaystyle r$ is also nonnegative) and $\displaystyle n \geq r$? what about if $\displaystyle r \geq 0$ and $\displaystyle n < r$? Remember, $\displaystyle r$ is an integer root of the polynomial, and $\displaystyle n$ is nonnegative.
    What about similar situations for when $\displaystyle r < 0$ (i.e. what are the possible cases if $\displaystyle r$ is negative?

    That should help you get started...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Aug 2012
    From
    NY
    Posts
    24
    Thanks
    1

    Re: Proving Divisibility

    I was thinking along the lines of proving by induction, but I'm not really sure.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Divisibility
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Oct 17th 2010, 06:43 PM
  2. Proving divisibility
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: May 11th 2010, 09:00 AM
  3. Divisibility
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Feb 27th 2010, 08:37 AM
  4. Proving an identity that's proving to be complex
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: Jul 21st 2009, 01:30 PM
  5. Proving divisibility question using contradiction
    Posted in the Discrete Math Forum
    Replies: 7
    Last Post: Mar 27th 2009, 11:00 PM

Search Tags


/mathhelpforum @mathhelpforum