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Math Help - Arghhhh Argument Principle

  1. #1
    Member Jason Bourne's Avatar
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    Arghhhh Argument Principle

    I get really annoyed when questions ask you to sketch stuff, like am doing maths and i'm supposed to be able to do art!

    Anyway, I'm not sure on this question:

    By sketching the graph of y = f(x) show that

    f(z) \equiv z^4 + 4z + 1 = 0

    has two negative real roots and no postitive real roots. Show also that f(z)=0 has no purely imaginary roots.

    Using the argument principle, determine in which quadrants of the complex plane the two complex roots lie.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Jason Bourne View Post
    I get really annoyed when questions ask you to sketch stuff, like am doing maths and i'm supposed to be able to do art!

    Anyway, I'm not sure on this question:

    By sketching the graph of y = f(x) show that

    f(z) \equiv z^4 + 4z + 1 = 0

    has two negative real roots and no postitive real roots. Show also that f(z)=0 has no purely imaginary roots.

    Using the argument principle, determine in which quadrants of the complex plane the two complex roots lie.
    Well, sketch the graph of y = x^4 + 4x + 1. You don't have to have an art degree, just get some graph paper and plug in some values of x and connect the dots with a smooth curve.

    You should be able to see that there are two negative and no positive roots.

    I don't know how to show that the other two roots are not pure imaginary by looking at the graph.

    -Dan
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    Quote Originally Posted by Jason Bourne View Post
    I get really annoyed when questions ask you to sketch stuff, like am doing maths and i'm supposed to be able to do art!

    Anyway, I'm not sure on this question:

    By sketching the graph of y = f(x) show that

    f(z) \equiv z^4 + 4z + 1 = 0

    has two negative real roots and no postitive real roots. Show also that f(z)=0 has no purely imaginary roots.

    Using the argument principle, determine in which quadrants of the complex plane the two complex roots lie.
    Suppose that z=ai for a\not = 0.
    Then f(ai) = a^4 +4ai+1=0 by equation real and imaginary parts we get (a^4+1)=0\mbox{ and }4a=0 which is impossible.

    What does this have to do with the principle of the argument?
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