Main question: Does anyone have any good/interesting applications of the rational root theorem or ways to teach it that don't involve conveniently ignoring computer-based tools in order to avoid rote checking of long root candidates lists?
Longer rationale: The rational root theorem feels like a redundant waste of time.
For example, the very first guided problem in my school's textbook (Holt, Algebra 2) demonstrates how to generate a candidate list, but then it immediately advises the student to use a graphing calculator to narrow down that list to the most likely roots they see on the graph in order to not have to plug and chug 10+ candidates. Why go through the trouble of generating that list if you're just going to look at the graph to identify the root? I get the point about verification, but why bother with the list at all if you trust the computer enough to believe its graph?
My students and I go through a few examples and practice problems and then check them on wolframalpha, but my students respond: "Why did just do all that work at the beginning?" I'm at a bit of a loss at what to say. This isn't like the quadratic formula, which not only is quick, but also there's clear motivation for not totally relying on computer-generated solutions (e.g., works for all quad polys; gives exact form of irrational and complex roots, not rounded). Moreover, it seems like an artificial chore to construct example polynomials with crazy enough rational roots (e.g., 12345/654321) to call into question the precision of computer-generated solutions. In other words, problems with numbers difficult enough to make us suspicious of calculator rounding are highly artificial and time-consuming to produce, as well as likely to require a computer to solve in a reasonable amount of time anyway without extreme drudgework.
I do see some drill-based skill-value in the repetitive process (factoring the constant term and leading coefficients and listing p/q for each, practice with evaluating a polynomial a various values), and I certainly think that discussions of what the theorem can and cannot tell you are important (e.g., if no rational roots, then the list doesn't apply). But more than many other topics of Algebra 2, this topic feels like its problems are far more artificially constructed, its solution process and methods are far less broadly illuminating, and the students are just going through the motions to get information that is more easily, more quickly, and more practically obtained in other ways.
[Update: 2/24] I generally agree with the responses saying that the RRT is a useful logical tool to help step the student up towards Calculus-level processes that require multiple systematic steps and the observation/juggling of multiple pieces of info in a given math expression.
I guess I'm just frustrated with the textbook presentation and style of problem I've seen up to now, which not only focuses on obtaining root values (rather than logically ruling things out), but also seems blandly circular in that you're often specifically constructing polynomials to have or not have specific rational roots (and for any appreciably high-degree polynomial example (10+), using a computer program to multiply it out). As noted in some response below, many polynomials that actually come up in practice are unlikely to have nice rational roots -- hence my concern about wasting time plugging and chugging after the candidate list has been generated.
I think I'm going to spend a bit of time writing problems more like the algebra puzzle in a comment below, i.e., provide the student with a finite set of options along with ancillary info which, when combined with the RRT, will rule out all but 1 or 2 of the options. This "tool for logic puzzle" approach seems much more fruitful and useful for stimulating abstract thinking than the "generate a list and try them all" presentation that practically begs you to use a computer to grind through rote calculations.
Thanks for all your comments!
[Update 2: 2/25]
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