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Help to solve a linear equation

  1. Dec 2, 2006 #1
    Hi!

    This is my first post at this forum. At the look of it I think i will be spending a lot of time here. I´ve have a problem that i need help with. Me and my boyfriend started a discussion yeasterday who was the smartest of us. So we gave each other a assignment. Mine was:

    Solve the equation

    | 1 1 1 1|
    | x 2 3 4|
    | x^2 4 9 16| = 0
    | x^3 8 27 64|

    I´ve have thought out that i need to use linear algebra to solve this. This is were my problem kick in. I´ve have just read moderate linaer algebra. So please can you help me. Don´t want to be the dumb one o:) We didn´t say how we should solve it...

    Thanks in advance! /X Sandra
     
  2. jcsd
  3. Dec 2, 2006 #2

    HallsofIvy

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    Expand the determinant using the first column:
    [tex]\left|\begin{array}{cccc}1 & 1 & 1 & 1 \\ x & 2 & 3 & 4 \\ x^2 & 4 & 9 & 16 \\ x^3 & 8 & 27 & 65 \end{array}\right|= [/tex]
    [tex]\left|\begin{array}{ccc}2 & 3 & 4 \\ 4 & 9 & 16 \\ 8 & 27 & 64\end{array}\right|[/tex]
    [tex]- x \left|\begin{array}{ccc}1 & 1 & 1 \\ 4 & 9 & 16 \\ 8 & 27 & 64\end{array}\right|[/tex]
    [tex]+ x^2 \left|\begin{array}{ccc}1 & 1 & 1 \\2 & 3 & 4 \\ 8 & 27 & 64\end{array}\right|[/tex]
    [tex]- x^3 \left|\begin{array}{ccc}1 & 1 & 1 \\2 & 3 & 4 \\ 4 & 9 & 16\end{array}\right|[/tex]

    Which gives a single cubic (not linear) equation to solve. (Hint: it has one integer solution.)
     
  4. Dec 2, 2006 #3
    Got it now. Thank you :!!)

    X/ Sandra
     
  5. Dec 3, 2006 #4

    HallsofIvy

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    And give your boyfriend a poke in the eye with a sharp stick!
     
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