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Fundamental theorem of algebra

  1. Nov 5, 2007 #1
    1. The problem statement, all variables and given/known data
    There are two versions of the fundamental theorem of algebra, one that says a polynomial of degree n has n roots and the other that says a polynomial can be factored into linear and irreducible quadratic factors. Is there a quick way to see how they are the same?


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 5, 2007 #2

    Gib Z

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    The 2nd version is talking within the real numbers ie A polynomial with real coefficients can be factored into linear and irreducible quadratic factors over R.

    The first version is the same because the factor theorem says that for P(x), if g is a root then (x-a) is a factor. We can use the quadratic formula too see that any quadratic factor can be factored into linear factors, if factored over C instead of R.
     
  4. Nov 5, 2007 #3
    I see why the second version implies the first version. I do not see why the first version implies the second version.

    How do you know that you can get rid of all of the factors (x - a) where a is complex, since the 2nd version really says that A polynomial with real coefficients can be factored into linear and irreducible quadratic with real coefficients.
     
  5. Nov 5, 2007 #4

    Gib Z

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    O yes I forgot about that implication. Remember the complex conjugate theorem, which states the for polynomials with real coefficients, complex roots will come in conjugate pairs.
     
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