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Homework Help: Factorization- any techniques ?

  1. Mar 22, 2010 #1
    1. The problem statement, all variables and given/known data
    I'm trying to factorize a characteristic equation from my ODEs class, and I am having a problem when it comes to 4th, 5th or higher order differential equations.
    Say this one:
    r5-3r4+3r3-3r2+2r=0


    2. Relevant equations



    3. The attempt at a solution

    Tried a lot of things but couldn't reach the result.
     
  2. jcsd
  3. Mar 22, 2010 #2

    ideasrule

    User Avatar
    Homework Helper

    There's no technique; you just guess and check, or plug the equation into your graphing calculator to find the roots. Have you tried r=1?
     
  4. Mar 22, 2010 #3

    Mark44

    Staff: Mentor

    Beg to differ with you ideasrule, at least as far as rational roots are concerned. The Rational Root Theorem says that if p/q is a solution of the equation anxn + ... + a1x1 + a0 = 0, the p has to divide a0 and q has to divide an.

    For the equation r5 - 3r4 + 3r3 -3r2 + 2r = 0,
    the left side can be factored, giving r(r4 - 3r3 + 3r2 -3r + 2) = 0.

    If the fourth degree factor has any rational roots, the possible candidates are +/-1 or +/-2. Each of these can be checked using either long division or synthetic division, which is essentially the same thing but a lot more efficient. As it turns out, using this theorem, the original fifth-degree polynomial can be factored into five linear factors, giving five solutions to the characteristic equation.
     
  5. Mar 24, 2010 #4
    Thank you guys for the replies. it was helpful...
     
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