Over the years I have occasionally encountered a number of Algebra 2 textbooks that make an incorrect (or at very least extremely misleading) claim along the lines that "all solutions of a polynomial equation can be found using a combination of the Rational Roots Theorem, Synthetic Division, and the Quadratic Formula." For example, the following image is from a McGraw-Hill Algebra 2 ebook:
I found a similar claim in the 2007 edition of Holt Algbra 2 (section 6-5, p. 441). While I haven't done an exhaustive curriculum search, I suspect this is fairly widespread. Even more to the point, I expect that many Algebra 2 teachers tell their students this, even if it is not written in the textbook.
It is of course true that the problems that students encounter in the course can all be handled by these methods, but that is because they have been carefully curated for that purpose. More generally, while the Fundamental Theorem of Algebra guarantees the existence of solutions, there are certainly polynomials of degree 3 and 4 that are irreducible over $\mathbb{Q}$, and finding zeros for such polynomials requires methods far beyond what is normally taught in high school. For degree 5 and higher, the Abel-Ruffini Theorem shows that it is in general not possible at all to find the irrational roots algebraically, and no "exact form" even exists, although the roots can be approximated to arbitrary precision using numerical methods.
Even when a text or teacher does not explicitly make the false claim that all polynomials can be solved using just a few methods, failing to state that in fact the opposite is true (while only presenting examples in which those methods work) seems to me to be at least a sin of omission, and I imagine most students come away believing that any polynomial equation can be solved.
Quite apart from the fact that we shouldn't be teaching students things that are actually false, the fact that some polynomial equations just can't be solved, no matter how clever you are, and that mathematics is capable of proving the impossibility of something, seems like an important piece of meta-knowledge to me. I put it in the same category as "you can't trisect an angle using only a compass and unmarked straightedge" -- we don't expect students to understand the proof, but the result seems worth knowing, if for no other reason than that it makes the constructions you can do seem more worthwhile.
My questions, after all this:
- How pervasive is this error in textbooks?
- Are there any high school textbooks that explicitly acknowledge that the methods included in the text are not adequate to solve all 3rd and 4th degree polynomial equations, and that in higher degrees that are no general methods at all?
- Has anybody ever studied this as an issue of teacher knowledge?