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Math Help - number of real roots?

  1. #1
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    number of real roots?

    The number of real solutions to x^4 - x^3 = cosec^2(x) - cot^2(x) is?

    -- ok, so i figured that cosec^2(x) = 1 + cot^2(x), so that brings it to
    x^4 - x^3 = 1 if i am right?

    so how many solutions are there then? 4?

    EDIT: using a graphics calc, i graphed x^4 - x^3 -1 =0 and checked how many x intercepts there were, and so there were 2 real solutions.
    But is there a way of doing this without the calculator?
    Last edited by scorpion007; October 23rd 2006 at 11:49 PM.
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  2. #2
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    Quote Originally Posted by scorpion007 View Post
    The number of real solutions to x^4 - x^3 = cosec^2(x) - cot^2(x) is?

    -- ok, so i figured that cosec^2(x) = 1 + cot^2(x), so that brings it to
    x^4 - x^3 = 1 if i am right?

    so how many solutions are there then? 4?

    EDIT: using a graphics calc, i graphed x^4 - x^3 -1 =0 and checked how many x intercepts there were, and so there were 2 real solutions.
    But is there a way of doing this without the calculator?
    Descartes' rule of signs tells us that x^4 - x^3 -1 =0 has exactly 1 positive
    root, and putting x=-y to get y^4+y^3-1=0, allows us to again apply the
    rule of signs to tell us that the original equation has exactly 1 negative
    root.

    So x^4 - x^3 -1 =0 has two real roots.

    (Descartes rule of signs may be found here. Note in this case it tells
    us exactly how many roots there are)

    RonL
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