14
votes
Are degrees of polynomials illogically defined in elementary algebra, intermediate algebra and college algebra courses?
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 ...
- 4,868
11
votes
Where do students learn to solve polynomial equations these days?
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 ...
- 7,215
9
votes
Non-US polynomial division notation
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 ...
- 211
9
votes
Non-US polynomial division notation
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. ...
- 191
9
votes
Accepted
Non-US polynomial division notation
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.
- 1,300
9
votes
Accepted
Where do students learn to solve polynomial equations these days?
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 ...
- 5,672
8
votes
Is the constant term a coefficient?
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'...
- 3,947
7
votes
Where do students learn to solve polynomial equations these days?
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 ...
6
votes
Are degrees of polynomials illogically defined in elementary algebra, intermediate algebra and college algebra courses?
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 ...
- 11.5k
6
votes
Definition of equation vs. expression vs. polynomial
Here are informal definitions of the terms that seem confusing to you:
A function is a relation between two sets, usually sets of numbers. It maps elements of the first set to elements of the second ...
- 28.9k
6
votes
Is coefficient same as constant?
I'd say that the video is not using the best word. I would call that constant the coefficient.
Constant means that it is a number and not a variable. That's true. But the word coefficient conveys more ...
- 19.2k
5
votes
Improving exposition of a proof about polynomials over infinite fields
How about this:
Let $k$ be an infinite field, and let $f \in k[x]$. Assume $f(t) = 0$ for all $t \in k$.
Assume to the contrary that $f$ is not the zero polynomial. Then $f$ is a polynomial of ...
- 23.2k
4
votes
Is the constant term a coefficient?
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 ...
- 3,011
4
votes
Is the constant term a coefficient?
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 ...
- 17.1k
4
votes
Where do students learn to solve polynomial equations these days?
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 ...
- 66
3
votes
Why is there variation in the meaning of "Standard form" for a quadratic?
When considering transformations of the "basic" $y=x^2$, there is not much else you can do besides
Vertical shift
Horizontal shift
Scaling
The first is achieved by adding a constant to $...
- 240
3
votes
Are degrees of polynomials illogically defined in elementary algebra, intermediate algebra and college algebra courses?
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 ...
- 1,060
3
votes
Where do students learn to solve polynomial equations these days?
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 ...
- 1,468
2
votes
Is the constant term a coefficient?
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 ...
- 8,846
2
votes
Are degrees of polynomials illogically defined in elementary algebra, intermediate algebra and college algebra courses?
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 ...
- 18.3k
1
vote
Is evaluating a Real Polynomial at a Complex Value a suitable task for Precalculus students?
Personally, I agree with your viewpoints on the topic. Apart from what you mention, such excercises do not unveil the reasons behind the emergence of complex numbers - which, at first glance, are a ...
- 1,060
1
vote
Are degrees of polynomials illogically defined in elementary algebra, intermediate algebra and college algebra courses?
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 $...
- 8,846
Only top scored, non community-wiki answers of a minimum length are eligible