Results 1 to 9 of 9
Like Tree2Thanks
  • 1 Post By greg1313
  • 1 Post By romsek

Thread: Polynomial problem

  1. #1
    Member
    Joined
    May 2010
    Posts
    79

    Polynomial problem

    Stuck on a problem...
    Can anyone help?

    I am after any polynomial with integer coeff that passes through the points ( 1,3) and (3,2).
    I started with quadratics and i think its not possible but i can't prove that either?

    Hint/help would be much appreciated!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    18,694
    Thanks
    2651

    Re: Polynomial problem

    Quote Originally Posted by rodders View Post
    Stuck on a problem...
    Can anyone help?

    I am after any polynomial with integer coeff that passes through the points ( 1,3) and (3,2).
    I started with quadratics and i think its not possible but i can't prove that either?

    Hint/help would be much appreciated!
    Two points determine a line (surely you learned that in geometry?)
    The graph of a linear equation is of the form y= ax+ b. Since it passes through the point (1, 3) we must have 3= a(1)+ b. Since it passes through (3, 2) we must have 2= a(3)+ b. Solve those two equation for a and b.


    You say "I started with quadratics and i think its not possible". It certainly is possible to find a parabola that passes through any two points- the problem is that there are an infinite number! We can write a quadratic as y= ax^2+ bx+ c. Since it is to pass through (1, 3) we must have 3= a(1)+b(1)+ c. Since it is to pass through (3, 2) we must have 2= 9a+ 3b+ c. That is only two equations with three unknowns. If we subtract a+ b+ c= 3 from 9a+ 3b+ c= 2 we get 8a+ 2b= 1. We can write that as b= (1/2)- 4a. Then a+ (1/2)- 4a+ c= -3a+ (1/2)+ c= 3. From that c= 3+ 3a- 1/2= 3a+ 5/2. Take a to be any number and those b and c will give a parabola passing through those two points. (If you take a= 0, you get the previous answer.)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    May 2010
    Posts
    79

    Re: Polynomial problem

    My problem is not trying to find any quadratic/polynomial but one with integer coefficients?
    Thats where i am stuck!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    6,420
    Thanks
    1676

    Re: Polynomial problem

    Hey rodders.

    Can you try starting by setting up a couple of congruence relationships for your polynomial?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    May 2010
    Posts
    79

    Re: Polynomial problem

    Not sure what you mean? Like Diophantine equations?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Dec 2016
    From
    Earth
    Posts
    54
    Thanks
    36

    Re: Polynomial problem

    Look at this pattern:

    (1, 3) & (3, 2)

    Start with a line:

    ax + b = f(x)

    3a + b = 2
    a + b = 3
    ------------
    2a = -1

    An even left-hand side equal to an odd integer means the variable a cannot be equal to an integer.


    Quadratic:

    ax^2 + bx + c = f(x)

    9a + 3b + c = 2
    a + b + c = 3
    ------------------
    8a + 2b = -1

    An even left-hand side equal to an odd integer means at least one of the variables cannot be an integer. *


    Cubic:


    ax^3 + bx^2 + cx + d = f(x)

    27a + 9b + 3c + d = 2
    a + b + c + d = 3
    --------------------------
    26a + 8b + 2c = -1

    See * above.


    And so on.

    After subtracting the two equations for any increasing degree polynomial, the left-hand side will always be an
    even-valued expression, while the right side equals 1.

    So, there is no such polynomial function with integer coefficients that passes through those two points.
    Last edited by greg1313; Dec 30th 2016 at 02:43 PM.
    Thanks from Archie
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Apr 2005
    Posts
    18,694
    Thanks
    2651

    Re: Polynomial problem

    Quote Originally Posted by rodders View Post
    My problem is not trying to find any quadratic/polynomial but one with integer coefficients?
    Thats where i am stuck!
    Do you understand that if you get an equation with fraction coefficients,, you can immediately get one with integer coefficients by multiplying the entire equation by the least common denominator of the fractions?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,258
    Thanks
    2238

    Re: Polynomial problem

    To make Greg's argument a bit clearer we can translate our curve so that the first point is at (0,0).

    This is a translation of (-1,-3) and applied to the second point we get (2,-1).

    Any polynomial without a constant term will pass through (0,0) so we are after

    $\displaystyle{\sum_{k=1}^n}~c_k 2^k = -1$

    $2\displaystyle{\sum_{k=1}^n}~c_k 2^{k-1} = -1$

    $\displaystyle{\sum_{k=1}^n}~c_k 2^{k-1} = -\dfrac 1 2$

    $c_k \in \mathbb{Z}$

    $2^{k-1} \geq 1 ,~k=1,\dots n \Rightarrow 2^{k-1} \in \mathbb{Z}$

    there is no way that the sum on the left ends up equaling a rational number as the integers are closed under multiplication and addition.

    thus there is no polynomial with integer coefficients that passes through these two points and thus no polynomial with integer coefficients that passes through the original two points.
    Last edited by romsek; Dec 30th 2016 at 10:18 PM.
    Thanks from rodders
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    May 2010
    Posts
    79

    Re: Polynomial problem

    Many thanks All! I sort of suspected this but i like the conclusive proof!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Problem #18: Polynomial
    Posted in the Math Challenge Problems Forum
    Replies: 5
    Last Post: Aug 17th 2014, 03:50 PM
  2. Polynomial problem
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: Jan 15th 2011, 03:44 AM
  3. [SOLVED] Polynomial problem
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Oct 8th 2010, 05:40 AM
  4. Polynomial problem
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Feb 3rd 2009, 02:45 PM
  5. A Problem of Polynomial
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Dec 9th 2008, 05:09 AM

/mathhelpforum @mathhelpforum