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Factoring Complex Functions

  1. Jan 18, 2010 #1
    1. The problem statement, all variables and given/known data

    The polynomial f(z) is defined by

    f(z) = z5 - 6z4 + 15z3 - 34z2 + 36z - 48

    Show that the equation f(z) = 0 has roots of the form z = ix where x is real, and hence factorize f(z)


    3. The attempt at a solution

    So I know that you begin by factoring out (z-ix) from the function, but I'm not quite sure how to work that out. I can only figure out how to get the first and last terms in the first step:

    f(z) = (z-ix)(z4 + ....... - 48i/x)

    How would you go about finding everything in between those two terms?
     
  2. jcsd
  3. Jan 18, 2010 #2
    The next term should be A*z^3, you have to find a.
    Now, we want a value A such that: when we expand the brackets, we get -6 for the coefficient of z^4.
    (z-ix)(z^4 + Az^3....)
    when you expand to get the z^4 term, we have: -ix*z^4 + A*z^4=-6*z^4.
    that means A=ix-6
    agree?
    now that you have A,
    can you do this for the rest of the terms?
    so what will you get?
     
  4. Jan 18, 2010 #3

    Mark44

    Staff: Mentor

    Evaluate f(ix) and then simplify all powers of i.
    Rewrite f(ix) as a complex number: g(x) + h(x)*i.
    Set f(ix) = 0. This implies that g(x) = 0 and h(x) = 0.
    Factor g(x) and h(x). This gives you a number of values of x for which f(ix) = 0.
     
  5. Feb 2, 2010 #4
    Thanks to both of you! I evaluated f(ix) and that simplified things quite a bit!
     
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