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Complex roots

  1. Aug 6, 2013 #1
    1. The problem statement, all variables and given/known data

    Hi so I'm not understanding my reading of this text.


    2. Relevant equations

    Below is what I don't understand.

    3. The attempt at a solution

    ## q_n(x) = (x-a_1)(x-a_2)(x-a_3)....(x-a_n) ##
    Now if we expanded this factorization by multiplying it out it should be clear the coefficient of ##x^n## should be one because it could only be derived by choosing x from each of the n bracket terms when composing the product.

    So what I don't understand what this is telling me. Expand the factorization.
    ## q_n(x) = (x-a_1)(x-a_2)(x-a_3)....(x-a_n) ##

    For the first three terms I got

    ##(-x^3a_3 + x^2a_2a_3 + x^2a_1a_3 - a_1a_2a_3x ) ...(x-a_n) ##



    Not sure how to include the nth term. How to write it.

    Would it be

    ##(a_3...a_nx^n - a_2a_3..a_nx^n -a_1a_3..a_nx^n + a_1a_2a_3..a_nx^n)##

    seems like crap. I don't get what this is telling me!!!
     
    Last edited by a moderator: Aug 6, 2013
  2. jcsd
  3. Aug 6, 2013 #2
    It is telling you that the term with [itex]x^n[/itex] after doing the multiplication comes with a factor of 1 because it is the result of multiplying the n [itex]x's[/itex] and all of them have a factor of 1! And [itex]1^n=1[/itex].

    Try with [itex]n=2[/itex] and then [itex]n=3[/itex].
     
  4. Aug 6, 2013 #3
    I get ##(x-a_1)(x-a_2)(x-a_3)\cdots(x-a_n)\\=(x^2-(a_1+a_2)x+a_1a_2)(x-a_3)\cdots(x-a_n)\\=(x^3-(a_1+a_2+a_3)x^2+(a_1a_2+a_2a_3+a_1a_3)x-a_1a_2a_3)\cdots(x-a_n).##
     
  5. Aug 6, 2013 #4
    How did you x^3 with no coefficent in front of it. I think you rearranged inbetween the steps.
    How did you get this.

    ##(x^3-(a_1+a_2+a_3)x^2+(a_1a_2+a_2a_3+a_1a_3)x-a_1a_2a_3)\cdots(x-a_n).##

    More steps. I see how you got the middle two parts but how do you have the last term before the nth as just coefficients?
     
  6. Aug 6, 2013 #5
    DOnt; answer this question thx cya
     
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