pain in neck calculus problem

From: Troubled (mkajumap_at_hotmail.com)
Date: 06/27/04 

Date: Sun, 27 Jun 2004 03:14:22 GMT

I need to find a 4th degree polynomials with 3 extrema and 2 points of
inflection, all 5 numbers being integers. I started off with letting y' = x
(x-p) (x-q) , with p and q integers (so the critical values will be 0,p,and
q). Then y" = (x-p)(x-q) +x(x-q) + x(x-p). I concluded that y"=0 when [(p+q)
+/- sqrt( (p+q)^2-3pq)]/3. I know that (p+q)^2-3pq must be a perfect
square, I^2. I concluded that p= [q +/-sqrt( (2I)^2 - 3q^2)]/2. This is
where I got stuck. At this point I'd pick values for q and I and see if p is
an integer and if it was I'd check to see if it satisfied the equation
above. But I can't find a p,q combo that works, nor do I like to randomly
pick numbers. Why is this problem so hard? Why can't I find a 4th degree
polynomial with any 5 integers (or any 5 numbers)that satisfy the statement?
Any hints or even solutions at this point would be appreciated. Believe it
or not this is not homework. It is the problem of the month at a college
which I do not attend that I came across.

Relevant Pages

  • Re: Algebraic integers
    ... that satisfy a polynomial of degree exactly n with coefficients in C, ... That's the point where allowing power series is different from simply ... allowing polynomials of unbounded degree. ...
    (sci.math)
  • Re: Algebraic integers
    ... So really we are dealing with a class of polynomials. ... "signs"(signs on there coefficients). ... that satisfy a polynomial of degree exactly n with coefficients in C, ... That's the point where allowing power series is different from simply ...
    (sci.math)
  • Re: 3 functional equations is max
    ... Take any positive integer n. ... Consider the polynomials ... Letting T f= f, ...
    (sci.math)
  • Re: Algebraic Generating Functions - Closure Properties Question
    ... Y+Z as a root and one that has Y*Z as a root, given polynomials for Y and Z. ... given the polynomials they satisfy), ... the resultant. ...
    (sci.math)
  • Re: a conjecture based on Euler pol. and the Little Fermat Theorem
    ... are there tomic polynomials that satisfy: ... it all comes back to the fact that the little fermat ... Simon Plouffe ...
    (sci.math)