13

I'm going to rewrite this answer to clarify what I think the issue is. I think the OP is imagining a different definition of the ring $k[x]$ than most answerers are. Here are two reasonable definitions: $k[x]$ is the ring of formal expressions of the form $\sum_{j=0}^{\infty} p_j x^j$ with $p_j \in k$ and we require that $p_j$ is $0$ for $j$ sufficiently ...


11

Students learn linear and quadratic equations in high school algebra. And then, if they have forgotten it, re-learn it in college, in courses called "pre-calculus" or something. Unless they specialize in mathematics at the college level, they do not learn any more. Why not? Because we have computers now, so most people do not need to solve polynomial ...


9

In Greece the students are taught polynomial division in the second class of upper high school (grade 11 at US educational system). It is the same algorithm as in Italy and Russia. Whole book in pdf form is available here.


9

In Italy, polynomial long division is usually presented as in the following example taken from one of the most widely used textbooks for first year high school students (M. Bergamini, G. Barozzi, A. Trifone, Matematica.blu 1, Zanichelli):


9

Here is a Russian 7th grade algebra textbook (publisher's website). Attached is a complete section dedicated to polynomial division, it is marked as optional.


9

The rational root theorem, synthetic division, the remainder theorem, Descartes rule of signs, and similar lower level topics were fairly widely taught in U.S. high school algebra-2 courses before and during the 1980s, but they've slowly been de-emphasized as graphing calculators (allows for numerical equation solving) came onto the scene at the end of the ...


8

Your question is kind of two parts: one about a convention Is the constant term a "coefficient" and one about a philosophy, which I perhaps find to be a more important question to answer. Isn't mathematics supposed to be non-arbitrary and consistent? Different fields of mathematics have different conventions; this can lead to some mathematicians ...


7

In the United States, solving linear and quadratic equations is a standard part of Algebra 1, which most students take in 8th or 9th grade. Students will return to polynomials and see long division and synthetic division in Algebra 2. This is also when students will learn the remainder theorem, Descartes’ Rule, and how to identify the only possible rational ...


6

This: "On the other hand, without the proof the definition of the degree of polynomials is not even logically established." is not quite right. What is needed to establish the definition is the fact that the degree is well-defined, but this fact is stated (implicitly) by stating the definition. The expectation that all facts be proven is out of place in a ...


4

It sometimes happens that slightly different definitions of the same word each have advantages and disadvantages. In such cases, I wouldn't be surprised to see some people supporting one definition and other people supporting a different definition. In the case at hand, though, I can't think of any advantages for defining "coefficient" to exclude the ...


4

I have to admit I was skeptical of the OP's claim that contemporary textbooks do not identify the constant term as a coefficient, so I checked the first book that I had handy -- and indeed it does seem to be the case, in at least my sample of 1. Here is some evidence: (Source: McDougal Littell Algebra 2, 2004.) Note however that 2004 precedes the Common ...


4

Synthetic division is a standard part of the stereotypical "algebra 2" course in the US (~grade 11) and is normally covered including drill problems and examination. In my experience, cubics and quartics are in the algebra 2 book (I just checked and they are in mine) but in sections with an asterisk. Typically the asterisk sections are not covered because ...


3

Most math majors never learn cardano's methods. As for synthetic division, it is part of most high school curricula and linear algebra courses. Cubic and quartic equations aren't part of any curriculum I'm aware of, not because of computers, but because students should learn "more important topics" and because the solutions of these equations are very long ...


3

Let's keep in mind that "mathematics" and "mathematics education" are different subjects. This question brings forth this distinction. At one end of the "Piaget" spectrum of mathematical stages of development, explaining that the constant term $c$ is the same as $cx^0$ might be too much cognitive overload. However, towards the other end of the spectrum, say ...


3

In general, I agree with @Henry Towsner on the fact that the proofs should not always be presented in an elementary course. However, I have to disagree on the implicit "well-definition property" of any definition. Such a definition would require that some sort of uniqueness property has been proved, which cannot - or should not - be done prior to the ...


2

I do not believe that this is a concern that would surface, or be worth surfacing, in the courses named by this question's title. In my reading, the question is analogous to worrying about whether you can ask about e.g. the thousands place of $7521$: To do so assumes that the base ten representation of $7521$ is unique, or, in particular, that "thousands ...


1

Soon after introducing polynomials, students learn to add and subtract polynomials. Notice that if $f(x)$ and $g(x)$ are polynomials with degrees $n$ and $m$ respectively, and if $f(x)-g(x)=0$, then $n=m$. It follows that the degree of a polynomial is well-defined. So the proof that degree is well-defined is not difficult at all. On the other hand, students ...


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