The function is...
I am familiar with how to use Descartes' Rule of Signs to determine how many possible positive/negative zeros there are. I am also familiar with how to use synthetic division to determine if a given value of x is a real zero.
The book I am working through is asking me to determine between which two integers the real zeros are located. When we divide a function, f(x) by x-r, the remainder basically tells us the value of f(r). If, when using synthetic division to divide by consecutive values of r (i.e. dividing a polynomial by x-1, x-2, x-3...) we notice that the sign of the remainder goes from positive to negative or vice-versa, we know that a zero occurs between these two integers since the remainder is equal to the value of f(r). This makes sense to me. What I don't understand, is how one knows what values of r to begin testing when the zeros are probably not rational.
The problem above is an example in the book I am working from. They created a chart and did simplified synthetic division to find the remainder for various values of r. They began with r=-3 aka x+3. Why did they not begin with -8? I ask this because the integral factors of the constant, 8, are and it seems like that would be a logical place to start considering you never really know from the get-go whether a function is going to have rational or irrational zeros.
Surely, some polynomials with a constant of 8 might have zeros that are < -3.
In any event, they observed two instances of the remainder changing signs between -2 and -1 AND 1 and 2. Since the degree is 3, that accounts for two of the three zeros. They also found a rational zero at x-4 aka x=4.
I ask about where to begin testing values of r because I am working on other problems and I am having a difficult time deciding where to begin. Do I start with r=-20 and work my way up until I observe the remainder changing signs enough times? (e.g. if degree is 4 then 4 sign changes). I suppose that technology would be a good starting point. Is there a way to determine what values of r to begin testing without using a graphing calculator or other technology?
Here's what is likely the origin of my confusion regarding the question "To use technology or to not use technology?" as it pertains to problems like these...
This is an example in my book. In this example x=unemployment rate (which I'm pretty sure can't be negative, since that would mean more people are employed than there are in the population). The book begins by saying "Now search for the zero of the related function." Then they show a simplified synthetic division table for r=17,18 which shows that the remainder changes from a positive to a negative from r equals 17 to 18. Then they say "There is a zero between 17 and 18 months." And then comes the part that confuses me, they say "Confirm this zero using a graphing calculator." The order in which they ask about and present the information makes it sound like I was only supposed to use the graphing calculator AFTER I determined there was a zero between r=17 and r=18. But, in order to determine that, I would probably use a graphing calculator in the first place right? (e.g. determine the zero's neighborhood by looking at the graph, table, etc.) This brings me back to my question above... is there a way to figure out a good starting place for testing out values of r when you suspect that the zeros are not going to be rational?
Perhaps the authors of the book meant for me to use technology from the beginning. If this is the case, I think it would be helpful if they always indicated when they expected the reader to use a graphing calculator. (side note: In some problems, they do actually say something like "graphing calculator problem"). Man, I should get a job as a text book checker... it may not be profitable for the publisher to pay someone like me, but it seems it would alleviate some of the uncertainty/confusion for at least the diligent students.
p.s. maybe the fact that they used the word "search" was a tip that i should use a graphing calculator... but even so, it is still somewhat vague.