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15 votes

How can we best motivate the study of polynomials to high-school students?

Using puzzles to attract attention: "Think of a number, subtract 7, multiply 3, add 30, divide by 3. Then subtract the original number. The result will always be 3. Why does this magic work?"...
Spai's user avatar
  • 299
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 ...
David E Speyer's user avatar
10 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 ...
Gerald Edgar's user avatar
  • 7,607
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 ...
Epameinondas's user avatar
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. ...
Luca Bressan's user avatar
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.
Rusty Core's user avatar
  • 1,317
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 ...
Dave L Renfro's user avatar
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'...
Opal E's user avatar
  • 4,043
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 ...
Tim Ricchuiti's user avatar
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 ...
Henry Towsner's user avatar
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 ...
Joseph O'Rourke's user avatar
6 votes

How can we best motivate the study of polynomials to high-school students?

If the student has ever used a vector-based computer drawing program, they will be familiar with Bézier curves. Bézier curves are extremely intuitive to understand for humans. They became popular as a ...
ComicSansMS's user avatar
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 ...
Sue VanHattum's user avatar
  • 21k
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 ...
Steven Gubkin's user avatar
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 ...
Andreas Blass's user avatar
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 ...
mweiss's user avatar
  • 17.4k
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 ...
guest's user avatar
  • 66
4 votes

How can we best motivate the study of polynomials to high-school students?

if you're willing to take a bit of diversion, you could show how polynomials are used for error correcting codes that make the internet possible (reed-Solomon, but do it with real numbers instead of ...
Oscar Smith's user avatar
4 votes

How can we best motivate the study of polynomials to high-school students?

I disagree with the claim by others that puzzles are more applicable to top students than struggling ones. Of course, you must choose the right puzzles for the right level, and actually teach them how ...
user21820's user avatar
  • 2,656
4 votes

How can we best motivate the study of polynomials to high-school students?

I think this question and the likely answers are heading you down a wrong path, because of unstated assumptions. You would be better off working with the student and helping him achieve. Baby steps ...
Post as a guest's user avatar
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 $...
Nij's user avatar
  • 229
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 ...
Vassilis Markos's user avatar
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 ...
user5402's user avatar
  • 1,528
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 ...
user52817's user avatar
  • 11k
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 ...
Benjamin Dickman's user avatar
1 vote

How can we best motivate the study of polynomials to high-school students?

Few ideas that come to mind: Let them predict trajectories of vehicles. Data can come from a simulator. Generate splines for scalable fonts targeting fictitious hieroglyphics
farhanhubble's user avatar
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 ...
Vassilis Markos's user avatar
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 $...
user52817's user avatar
  • 11k

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