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Math Help - Polynomial and Rational inequalities help

  1. #1
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    Polynomial and Rational inequalities help

    I have a couple problems I dont understand any help is appreciated..

    1.) x^4 - 3x^2 - 4 > 0 solve algebraically.

    2.) degree 4: zeros: 3+2i; 4, mulpilicity 2
    3.) degree 4: zeros: 3, multiplicity 2; -i

    I dont understand 2 and 3 at all, if anyone has any outside sources or good definitions for the problems it would be nice

    thanks
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  2. #2
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    I'll try to explain this, but it's hard to explain without knowing how much you know already.

    For 2 and 3, the degree is the highest power of x in the equation, so x^5 would have degree 5, and x^5 + x -1 would still have degree 5.

    Multiplicity would mean how many times a certain value is makes the equation equal to 0, so x^2 would have zeroes at 0 with multiplicity 2, whereas x would only have it with multiplicity 1. you can think of it as being able to factor out zeroes, so for x^2 it's the same as x*x, so there's two ways to get 0.

    I'm assuming that you want real valued polynomials, so that means that for every complex number you get, it's conjugate is also a zero of the function. A complex number takes the form x + iy, and it's conjugate is just x - iy.

    2.) degree 4: zeros: 3+2i; 4, mulpilicity 2

    This means that you have 4 different factors from the degree, so you want to multiply together 4 things that have only degree equal to 1, so it's (x + a) (x+b)(x+c)(x+d) where a, b, c, d are all some value. Now we know that it has to zeroes at 4, so when you plug in 4 you get a 0, which only happens for (x-4). Since the multiplicity is 2, you have this two times. Te other zero is 3+2i, and it's conjugate must also be a zero, so you have 3+2i and 3-2i, which achieve zero at (x-3-2i) and (x-3+2i). Multiply these together: (x-4)(x-4)(x-3-2i)(x-3+2i) and you get your answer.
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  3. #3
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    Talking

    Quote Originally Posted by cokeclassic View Post
    I dont understand 2 and 3 at all, if anyone has any outside sources or good definitions for the problems it would be nice
    Try reviewing some online lessons regarding how to find quadratics and polynomials from their zeroes.
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