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Nature of the roots of the equations

  1. Dec 23, 2013 #1
    1. The problem statement, all variables and given/known data



    2. Relevant equations


    The given equations are;

    Q.1) Find the no of + roots of the equation x^4 -4x+1=0

    Q.2) Find the no. of negative roots of the eqn x^4-4x-1=0

    Q.3) Find the no of complex roots of the eqn x^4-4x-1=0

    3. The attempt at a solution
     
  2. jcsd
  3. Dec 23, 2013 #2

    SteamKing

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  4. Dec 23, 2013 #3
    yes indeed but according to our maths teacher it's pretty rubbish in practical use because it just gives the MAXIMUM possible with any equation no matter the respective conditions..


    no help i guess from that :(
     
  5. Dec 23, 2013 #4
    Have you learned about derivatives yet? This problem seems really well suited for a first and second derivative test and after a little argument you can answer the question.

    I'm not sure if this is how your teacher wants it completed, I forget if derivatives are covered in pre-calculus, it's been awhile since I looked at a book.
     
  6. Dec 23, 2013 #5

    SteamKing

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    The Rule of Signs is but one tool in your previously empty toolbag.

    You should know how many roots these equations will have. By careful use of this fact and the Rule of Signs, it may help you to narrow the numbers of each type of root.
     
  7. Dec 23, 2013 #6

    haruspex

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    I think you'll find it has some value here. (And please clarify whether the three equations are supposed to be the same. As posted, they're not.)
     
  8. Dec 23, 2013 #7

    SammyS

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    Assuming that all three questions are regarding the same equation, then you are being asked about the zeros of one of the two following polynomials.

    ##\displaystyle P(x)= x^4 -4x+1 ##

    OR

    ##\displaystyle Q(x)=x^4 -4x-1 ##

    For either case, Descartes' Rule of Signs turns out to be very useful when combined with some other observations.

    Maybe it's not rubbish for these polynomials.
     
  9. Dec 24, 2013 #8
    Yes i know a good deal on how to do the derivatives now after a year.

    but these three separate questions were part of a question bank on the chapter QUADRATIC EQUATIONS.

    So i guess calculus goes out of the window for now.

    The only way to solve this is by discarte? and ofc derivative
     
  10. Dec 24, 2013 #9

    SammyS

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    Well, you haven't resolved the question regarding which function (or equation) is being asked about.

    If you're being asked about the roots of x4-4x-1 = 0 , then Descartes along with a nominal amount of knowledge about the zeros of polynomials, will give all the answers. While Descartes' Rule gives the maximum number of positive roots or maximum number of negative roots, if the number is fewer than either maximum, if the number of either is less than the maximum, then the amount that it is less is an even number. So for this equation there is one positive root and one negative root. How many roots does a degree 4 polynomial equation have?

    ...
     
  11. Dec 24, 2013 #10

    berkeman

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    Please check your PMs. It is against the rules of these forums to post a schoolwork question without showing any effort toward the solution. It is also against the PF rules for other posters to offer help when you show zero effort in your first post in a thread.
     
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