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Math Help - polynomial powers and root quatity

  1. #1
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    polynomial powers and root quatity

    Hi;
    I thought that if I had a 4degree polynomial it has 4 roots,
    a 5degree polynomial has 5 roots ect...


    But I'm just starting to read an article on Descartes' rule of signs and it says
    you can dertermine the amount of roots a polynomial can have using this rule,

    why have the rule if we can tell from the power?
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  2. #2
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    Re: polynomial powers and root quatity

    Quote Originally Posted by anthonye View Post
    I thought that if I had a 4degree polynomial it has 4 roots,
    a 5degree polynomial has 5 roots ect...
    This is true (by the fundamental theorem of algebra) only if you allow complex roots. The polynomial x⁴ + 1 has no real roots because x⁴ ≥ 0 for all x.

    Besides, Descartes' rule of signs allows estimating the number of positive roots. For example, (x + 1)(x - 1) has two roots but only one positive root.
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    Re: polynomial powers and root quatity

    So if I account for complex roots the higest power is the number of roots?

    Descartes' only allows for positive roots ?
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    Re: polynomial powers and root quatity

    Quote Originally Posted by anthonye View Post
    So if I account for complex roots the higest power is the number of roots?

    Descartes' only allows for positive roots ?
    Descartes' Rule of Signs
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    Re: polynomial powers and root quatity

    DesCarte's rule of signs says that the number of positive real roots of a polynomial is either equal to the number of changes of sign or less than that by a multiple of two. To deal with negative roots, replace "x" with "-x".

    For example, x^5+ 4x^4- x^3+ x^2+ x- 5 has 3 changes of sign: from positive to negative, at -x^3, back to positive at + x^2 and negative again at -5. That tells us that there are either 3 or 1 postive roots. If we replace x with -x, the odd powers of x change sign giving -x^5+ 4x^4+ x^3+ x^2- x- 5 which has 2 changes in sign and so either 2 or 0 negative roots. Of course, that does NOT say anything about complex roots. DesCarte's rule of signs tells us there may be
    1) 3 positive, 2 negative, and 0 complex roots
    2) 1 positive, 2 negative, and 2 complex roots
    3) 3 positive, 0 negative, and 0 complex roots
    4) 1 positive, 0 negative, and 4 complex roots
    There have to be an even number of complex roots, of course, because, since the coefficients are real numbers, the complex roots come in complex conjugate pairs.
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