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Math Help - Show an equation has at most a certain amount of roots

  1. #1
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    Show an equation has at most a certain amount of roots

    Dont know how to answer the more general forms of these questions using Rolle's Theorem and mean value theorem, for example;

    - Show that the equation x^3-15x+c=0 has at most one root in the interval  [-2,2]
    - Show that a polynomial of degree 3 has at most three real roots
    - Show that a polynomial of degree n has at most n real roots.
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by Robb View Post
    Dont know how to answer the more general forms of these questions using Rolle's Theorem and mean value theorem, for example;

    - Show that the equation x^3-15x+c=0 has at most one root in the interval  [-2,2]
    - Show that a polynomial of degree 3 has at most three real roots
    - Show that a polynomial of degree n has at most n real roots.


    First one:

    Suppose it has more than one, for example, 2. Then the mean value theorem says that there is a number in the interior of [-2, 2] where the first derivative is 0. But f'(x) = 3x^2 - 15 = 3 (x^2 - 5), and its roots are clearly outside the interval.

    The other two problems are similar. The result you're using here is called Rolle's Theorem, a special case of the Mean Value Theorem.

    Good luck!!
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