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Math Help - Finding Rational Zeros: Methods

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    Finding Rational Zeros: Methods

    Alright, these can't be the only ways...

    I use the theorem and get all these possible rational zeros. And I could go through them one by one seeing if one of them works out and yeah, there's like a list of 20 of them (but really 40) I have to sift through hoping I get lucky. And so the ones I try out I use upper and lower bounds to try and enhance my search by restricting what I search for to narrow in on it. Even this though is time consuming. And I'm not allowed to use a graphing calculator. I have to do this at least within 5 minutes because my quizzes are timed.

    Did anyone invent a better method that's not in the textbook that I can use to get these done quickly? Time is of the essence.
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    Quote Originally Posted by A Beautiful Mind View Post
    Alright, these can't be the only ways...

    I use the theorem and get all these possible rational zeros. And I could go through them one by one seeing if one of them works out and yeah, there's like a list of 20 of them (but really 40) I have to sift through hoping I get lucky. And so the ones I try out I use upper and lower bounds to try and enhance my search by restricting what I search for to narrow in on it. Even this though is time consuming. And I'm not allowed to use a graphing calculator. I have to do this at least within 5 minutes because my quizzes are timed.

    Did anyone invent a better method that's not in the textbook that I can use to get these done quickly? Time is of the essence.
    Here is one trick if you know calculus:

    if you have a polynomial of odd degree and the derivative is always increasing, then there is at most one real root. Then after you see one rational root you dont need to continue to check the others.

    Otherwise: if you find a rational root try to factor it out of the original polynomial. for instance if 1 is a root then factor out (x-1). Perhaps this leaves you with a quadratic or something where you can see the roots quickly.


    In general: i dont think so :O(
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    Quote Originally Posted by robeuler View Post
    Here is one trick if you know calculus:

    if you have a polynomial of odd degree and the derivative is always increasing, then there is at most one real root. Then after you see one rational root you dont need to continue to check the others.

    Otherwise: if you find a rational root try to factor it out of the original polynomial. for instance if 1 is a root then factor out (x-1). Perhaps this leaves you with a quadratic or something where you can see the roots quickly.


    In general: i dont think so :O(
    I don't know Calculus yet, but I guess I could learn about derivatives if this makes it the slightest bit easier.

    I wonder why though this hasn't been worked on...or maybe it has and there's truly no other way. Ugh.
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    Quote Originally Posted by A Beautiful Mind View Post
    I don't know Calculus yet, but I guess I could learn about derivatives if this makes it the slightest bit easier.

    I wonder why though this hasn't been worked on...or maybe it has and there's truly no other way. Ugh.
    It was proved by Abel (I think) that there is, in general, no systematic way of solving a polynomial of order 5 "by radicals" (whatever that means, i.e. by using elementary algebra). If you get given something of high-order to solve, then you can guarantee there's going to be some straightforward factorisation that can be done - and in that case, listing all possibilities and examining them is usually the only practical way of doing it until you get a "feel" for it.
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