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Find a >2 degree polynomial that satisfies (1, p), (2, q), (3, r) - p,q,r = arbitrary

  1. Sep 8, 2006 #1
    Find a polynomial of degree < 2 [a polynomial of the form [itex]f\left(t\right)\,=\,a\,+\,bt\,+\,ct^2[/itex]] whose graph goes through the points (1, p), (2, q), (3, r), where p, q, r are arbitrary constants. Does such a polynomial exist for all values of p, q, r?


    Here is what I have so far:

    (1, p) ==> [itex]p\,=\,a\,+\,b\,+\,c[/itex]
    (2, q) ==> [itex]q\,=\,a\,+\,2b\,+\,4c[/itex]
    (3, r) ==> [itex]r\,=\,a\,+\,3b\,+\,9c[/itex]

    [tex]\begin{displaymath}
    \left[ \begin{array}{ccc|c}
    1 & 1 & 1 & p \\
    1 & 2 & 4 & q \\
    1 & 3 & 9 & r
    \end{array} \right]
    \end{displaymath}[/tex]

    [tex]R_2\,=\,R_2\,-\,R_1[/tex]
    [tex]R_3\,=\,R_3\,-\,R_1[/tex]

    [tex]\begin{displaymath}
    \left[ \begin{array}{ccc|c}
    1 & 1 & 1 & p \\
    0 & 1 & 3 & q - p \\
    0 & 2 & 8 & r - p
    \end{array} \right]
    \end{displaymath}[/tex]

    [tex]R_3\,=\,R_3\,-\,2R_2[/tex]

    [tex]\begin{displaymath}
    \left[ \begin{array}{ccc|c}
    1 & 1 & 1 & p \\
    0 & 1 & 3 & q - p \\
    0 & 0 & 2 & r - q
    \end{array} \right]
    \end{displaymath}[/tex]

    [tex]R_3\,=\frac{1}{2}\,R_3[/tex]

    [tex]\begin{displaymath}
    \left[ \begin{array}{ccc|c}
    1 & 1 & 1 & p \\
    0 & 1 & 3 & q - p \\
    0 & 0 & 1 & \frac{1}{2}\left(r - q\right)
    \end{array} \right]
    \end{displaymath}[/tex]

    [tex]R_2\,=\,R_2\,-\,3R_3[/tex]

    [tex]\begin{displaymath}
    \left[ \begin{array}{ccc|c}
    1 & 1 & 1 & p \\
    0 & 1 & 0 & \frac{5}{2}\,q\,-\,p\,-\,\frac{3}{2}\,r \\
    0 & 0 & 1 & \frac{1}{2}\left(r - q\right)
    \end{array} \right]
    \end{displaymath}[/tex]

    [tex]R_1\,=\,R_1\,-\,R_2\,-\,R_3[/tex]

    [tex]\begin{displaymath}
    \left[ \begin{array}{ccc|c}
    1 & 0 & 0 & 2p\,-\,2q\,+\,r \\
    0 & 1 & 0 & \frac{1}{2}\left(5q\,-\,2p\,-\,3r\right) \\
    0 & 0 & 1 & \frac{1}{2}\left(r - q\right)
    \end{array} \right]
    \end{displaymath}[/tex]

    [tex]a\,=\,2p\,-\,2q\,+\,r[/tex]
    [tex]b\,=\,\frac{1}{2}\,\left(5q\,-\,2p\,-\,3r\right)[/tex]
    [tex]c\,=\,\frac{1}{2}\,\left(r\,-\,q\right)[/tex]


    [tex]f\left(t\right)\,=\,a\,+\,bt\,+\,ct^2[/tex]
    [tex]f\left(t\right)\,=\,\left(2p\,-\,2q\,+\,r\right)\,+\,\left[\frac{1}{2}\,\left(5q\,-\,2p\,-\,3r\right)\right]\,t\,+\,\left[\frac{1}{2}\,\left(r\,-\,q\right)\right]\,t^2[/tex]
    [tex]f\left(t\right)\,=\frac{1}{2}\,r\,t^2\,-\,\frac{1}{2}\,q\,t^2\,+\,\frac{5}{2}\,q\,t\,-\,\frac{3}{2}\,r\,t\,-\,p\,t\,+\,2\,p\,+\,r\,-\,2\,q[/tex]

    But when I try a set of arbitrary numbers [(1, 8), (2, 12), (3, 3)], I get an equation that does not match the points:

    [tex]f\left(t\right)\,=\,-\frac{9}{2}\,t^2\,+\,\frac{35}{2}\,t\,-\,5[/tex]

    [tex]f\left(1\right)\,=\,-\frac{9}{2}\,(1)^2\,+\,\frac{35}{2}\,(1)\,-\,5\,=\,8[/tex]

    [tex]f\left(2\right)\,=\,-\frac{9}{2}\,(2)^2\,+\,\frac{35}{2}\,(2)\,-\,5\,=\,12[/tex]

    [tex]f\left(3\right)\,=\,-\frac{9}{2}\,(3)^2\,+\,\frac{35}{2}\,(3)\,-\,5\,=\,7[/tex]

    However, [itex]f\left(3\right)[/itex] should be equal to r, which is 3, not 7.

    Please help:)
     
    Last edited: Sep 8, 2006
  2. jcsd
  3. Sep 8, 2006 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Of degree < 2? Then it is linear, of the form y= mx+ b and you can only determine only the two numberss m and b. In general 2 equations will do that. You can choose m and b so that 2 of the requirements (1, p), (2, q), (3, r) are satisfied but not all 3.
     
  4. Sep 8, 2006 #3

    radou

    User Avatar
    Homework Helper

    For (p, q, r) = (0, 0, 0) we have (a, b, c) = (0, 0, 0), and therefore a zero polynomial, which has a degree undefined or [tex]-\infty[/tex], so it's less than 2. :biggrin:
     
  5. Sep 8, 2006 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    The title says "Find a >2 degree polynomial" and the post itself says
    "Find a polynomial of degree < 2".

    Is it any wonder I'm confused!
     
  6. Sep 8, 2006 #5

    Mute

    User Avatar
    Homework Helper

    Confusion about <2 vs >2 aside, your derivation has an error in the quoted section. You substract [itex]2 R_2[/itex] from [itex]R_3[/itex], but in the augmented part of the matrix you've only subtracted [itex]1 R_2[/itex], resulting in r - q instead of r + p - 2q. I didn't check to see if everything else was correct, but this is certainly a place to start with.
     
  7. Sep 11, 2006 #6
    Sorry for the confusion!

    The polynomial needs to have a degree of GREATER THAN OR EQUAL TO 2.

    So it needs to have at least one squared term.

    Here is the corrected derivation with help from Mute (Thanks!):

    (1, p) ==> [itex]p\,=\,a\,+\,b\,+\,c[/itex]
    (2, q) ==> [itex]q\,=\,a\,+\,2b\,+\,4c[/itex]
    (3, r) ==> [itex]r\,=\,a\,+\,3b\,+\,9c[/itex]

    [tex]\begin{displaymath}
    \left[ \begin{array}{ccc|c}
    1 & 1 & 1 & p \\
    1 & 2 & 4 & q \\
    1 & 3 & 9 & r
    \end{array} \right]
    \end{displaymath}[/tex]

    [tex]R_2\,=\,R_2\,-\,R_1[/tex]
    [tex]R_3\,=\,R_3\,-\,R_1[/tex]

    [tex]\begin{displaymath}
    \left[ \begin{array}{ccc|c}
    1 & 1 & 1 & p \\
    0 & 1 & 3 & q - p \\
    0 & 2 & 8 & r - p
    \end{array} \right]
    \end{displaymath}[/tex]

    [tex]R_3\,=\,R_3\,-\,2R_2[/tex]

    [tex]\begin{displaymath}
    \left[ \begin{array}{ccc|c}
    1 & 1 & 1 & p \\
    0 & 1 & 3 & q - p \\
    0 & 0 & 2 & r - 2q + p
    \end{array} \right]
    \end{displaymath}[/tex]

    [tex]R_3\,=\frac{1}{2}\,R_3[/tex]

    [tex]\begin{displaymath}
    \left[ \begin{array}{ccc|c}
    1 & 1 & 1 & p \\
    0 & 1 & 3 & q - p \\
    0 & 0 & 1 & \frac{1}{2} r - q + \frac{1}{2} p
    \end{array} \right]
    \end{displaymath}[/tex]

    [tex]R_2\,=\,R_2\,-\,3R_3[/tex]

    [tex]\begin{displaymath}
    \left[ \begin{array}{ccc|c}
    1 & 1 & 1 & p \\
    0 & 1 & 0 & -\frac{5}{2}\,p\,+\,4\,q\,-\,\frac{3}{2}\,r \\
    0 & 0 & 1 & \frac{1}{2} r - q + \frac{1}{2} p
    \end{array} \right]
    \end{displaymath}[/tex]

    [tex]R_1\,=\,R_1\,-\,R_2\,-\,R_3[/tex]

    [tex]\begin{displaymath}
    \left[ \begin{array}{ccc|c}
    1 & 0 & 0 & 3\,p\,-\,3\,q\,+\,r \\
    0 & 1 & 0 & -\frac{5}{2}\,p\,+\,4\,q\,-\,\frac{3}{2}\,r \\
    0 & 0 & 1 & \frac{1}{2} r - q + \frac{1}{2} p
    \end{array} \right]
    \end{displaymath}[/tex]

    [tex]a\,=\,3\,p\,-\,3\,q\,+\,r[/tex]
    [tex]b\,=\,-\frac{5}{2}\,p\,+\,4\,q\,-\,\frac{3}{2}\,r[/tex]
    [tex]c\,=\,\frac{1}{2} r - q + \frac{1}{2} p[/tex]


    [tex]f\left(t\right)\,=\,a\,+\,bt\,+\,ct^2[/tex]
    [tex]f\left(t\right)\,=\,\left(3\,p\,-\,3\,q\,+\,r\right)\,+\,\left[-\frac{5}{2}\,p\,+\,4\,q\,-\,\frac{3}{2}\,r\right]\,t\,+\,\left[\frac{1}{2} r - q + \frac{1}{2} p\right]\,t^2[/tex]
    [tex]f\left(t\right)\,=\frac{1}{2}\,r\,t^2\,+\,\frac{1}{2}\,p\,t^2\,-\,q\,t^2\,+\,4\,q\,t\,-\,\frac{5}{2}\,p\,t\,-\,\frac{3}{2}\,r\,t\,+\,3\,p\,+\,r\,-\,3\,q[/tex]

    [tex]p\,=\,8[/tex]
    [tex]q\,=\,12[/tex]
    [tex]r\,=\,3[/tex]

    So now it works for these numbers!

    [tex]f\,(\,t\,)\,=\,-\frac{13}{2}\,t^2\,+\,\frac{47}{2}\,t\,-\,9[/tex]

    [tex]f\,(\,1\,)\,=\,-\frac{13}{2}\,(\,1\,)^2\,+\,\frac{47}{2}\,(\,1\,)\,-\,9\,=\,8[/tex]
    [tex]f\,(\,2\,)\,=\,-\frac{13}{2}\,(\,2\,)^2\,+\,\frac{47}{2}\,(\,2\,)\,-\,9\,=\,12[/tex]
    [tex]f\,(\,3\,)\,=\,-\frac{13}{2}\,(\,3\,)^2\,+\,\frac{47}{2}\,(\,3\,)\,-\,9\,=\,3[/tex]

    But how can I figure (or prove) whether or not a polynomial exists for all values of p, q, and r? There does exist such a polynomial for all p, q, and r right?
     
    Last edited: Sep 11, 2006
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