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Factoring equation with real coefficients

  1. Mar 18, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the roots of [tex] z^4+4=0 [/tex] and use that to factor the expression into quadratic factors with real coefficients.

    2. Relevant equations
    DeMoivre's formula.

    3. The attempt at a solution
    I have been able to identify they are [tex] \pm 1 \pm i [/tex] but i have no idea how to factor the expression.
    Thanks!!
     
  2. jcsd
  3. Mar 18, 2015 #2

    Dick

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    No, they aren't. None of those are roots. Try them. Use deMoivre!
     
  4. Mar 19, 2015 #3
    [tex] (1+i)^4+4=0 [/tex]
    [tex] (-1+i)^4+4=0 [/tex]
    [tex] (1-i)^4+4=0 [/tex]
    [tex] (-1-i)^4+4=0 [/tex]

    Not sure what you're talking about.
     
  5. Mar 19, 2015 #4

    SammyS

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    It is possible to do this in the reverse order. That is:
    1. Factor ##\ z^4+4\ ## into quadratic factors with real coefficients.
    then
    2. find the roots of ##\ z^4+4=0\ .\ ##​

    That's not following the instructions, but it may give some insight.

    Suppose ##\ z^4+4=(z^2+az+2)(z^2+bz+2)\ ##.
    Expand the right hand side & equate coefficients.
     
  6. Mar 19, 2015 #5

    Dick

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    Sorry! I read your post as saying the roots were ##\pm 1## and ##\pm i##. If ##r_1## and ##r_2## are roots then ##(z-r_1)(z-r_2)## is a factor of your polynomial. Try multiplying that out when ##r_1## and ##r_2## are complex conjugates.
     
  7. Mar 19, 2015 #6

    HallsofIvy

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    Yes, 1+ i, 1- i, -1- i, and -1+ i are roots so the we can write [itex]z^4+ 4= (z- (1+ i))(z- (1- i))(z- (-1+ i)(z- (-1- i))= (z- 1- i)(z- 1+ i)(z+ 1- i)(z+ 1+ i)[/itex]
    Write [itex](z- 1- i)(z- 1+ i)= ((z- 1)- i)((z- 1)+ i)[/itex] and [itex](z+ 1- i)(z+ 1+ i)= ((z+ 1)- i)((z+ 1)+ i)[/itex]. Now use the fact that [itex](a- b)(a+ b)= a^2- b^2[/itex].
     
  8. Mar 19, 2015 #7
    Thanks a lot this worked great!
     
  9. Mar 19, 2015 #8

    SammyS

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    What did you get for the quadratic factors?
     
  10. Mar 19, 2015 #9
    [tex](z^2-2z+2)(z^2+2z+2)[/tex]
     
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