# Quickly identifying upper and lower bounds for zeros

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I have a quick question regarding the identification of the lowest upper bound and the greatest lower bound for the zeros of a polynomial function. The method that my book shows is to use to upper and lower bound theorem involving synthetic division. This involves performing division with 1, 2, 3, etc. and -1, -2, -3 etc. until the bounds are found, but what if the lowest upper bound is something like 57 or 89? I'm I supposed to perform division for every number or is there a way to identify a smaller group of possible bounds.

I've just been using the results of division to estimate about where it would be and eliminated possibilities from there, it just does not seem very...I don't know.. math-like to have to do that.

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There's not always an elegant and simple solution to every problem that you come across. Lots of the time an algorithm is messy, expensive to run and painfully tedious to do by hand.

For the purposes of school, no-one is going to asses you on hours of long division, so you can assume that the lowest upper bound probably wont be 89.

Without that perspective, there might be something you can do to improve upon your algorithm.

In this case, I don't think there's a tool to replace the one your using. But you could possibly contain the amount of work that you're doing with Sturm's theorem.

On top of that, you'd have to use a certain degree of common sense. Rather than trying 1,2,3 and 4, you might want to try 1,10,100,1000 and then if you get a result, begin narrowing it down further until your find the lowest upper bound. (of course, at that point you're talking about optimising a computer program, not anything that you would ever bother doing by hand).

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Thank you for your response. I'll be looking into Sturm's theorem, although it's probably way over my head at this point.

I've pretty much just been doing what you mentioned. I just look at a specific term and estimate about what I need to change my divisor to in order to make it positive or negative. I was just wondering if there was something that I missed. I suppose that i'll just have to get used to the idea that not everything is perfectly clear, especially as I progress into more advanced mathematics.

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