Not dead wood for sure, and for sure more interesting than long division. In school we learned all three methods, and it showed us the beauty and elegance of algebraic optimizations.
You found it interesting and informative, but the vast majority of students would find it merely another thing to rote-memorise if they can until the quiz and hope it’s not on the final.
The vast majority of those students will never solve a quadratic equation again once they get out of school. In the very few times they may need to directly use one they will use an online calculator or an app instead.
Any time in quadratic equations not spent on their practical use and application is wasted for most students. The most important thing that they need to take away is that that class of problems can be identified and have definite solutions, and how to intuit if a problem falls into that category.
Save the synthetic division for the AP or other optional classes, where people’s passions will lead them and they will benefit because they have self selected to benefit. The people that might benefit from the insights given will be there.
For everyone else critical thinking and problem solving are far, far higher priorities.
What is needed is to abandon the one size fits all approach to education. It harms the highest academic potential students, as well as the ones whose interests and capabilities lie in other areas.
Although I agree with the slant of your argument, that pedagogy should emphasize application over abstraction especially in foundational survey classes in a given subject, synthetic division is a rather feeble boogeyman to beat up on.
It's a hack almost on the level of "one weird trick" complexity that saves time when you need to test a polynomial for certain types of results, and because it's more of a trick than a technique it takes all of about 5 minutes to demonstrate, and probably multiplies out to 15 minutes to practice to mastery for the sake of a quiz.
On the other hand it could probably be swapped out for a greater emphasis on a technique that appears equally obscure at the time you're learning it, partial fraction decomposition, because even though like synthetic division it will never be useful on its own it becomes a critical tool just a little later in the math curriculum.
But beyond all of this a basic paradox of teaching math beyond the most elementary level of basic algebra and statistics, and perhaps geometry, is that much of what is learned beyond these very fundamental concepts does not have an application in "real life" in a meaningful way.
The application of much of this early-intermediate material is solely as a toolchain for dealing with more advanced kinds of math, and there is no way to tell to what extent any given student will be able to leverage these tools depending on what aspects of math they are interested in developing further, or if they're interested in developing in that direction at all.
Yeah, it’s a difficult problem. My preference would be to treat basic algebra with a few weeks of very basic applied geometry and trigonometry as the baseline goal. Anything more advanced that should be elective, and there should also be a trades math track that focuses on applying those basic skills to practical problems as well.
But we really need mandatory critical thinking and problem solving, with a little bit of practical statistics that focuses on interpretation of statistical figures in the wild.
