[QUOTE=R.D. Silverman;218598]But you keep claiming that you are ready for more advanced material!!
(calculus, college algebra, etc).

The polynomial exercizes are typical of ones that I encountered when taking
pre-calculus in high school. If you can not handle them, then you are
not, IMO, ready for more advanced material.

Would anyone else like to render an opinion?

The exercizes I gave should be typical for a high level honors high
school class.[/QUOTE]

Since we haven't really seen the level of material with which Tomer [I]is[/I] comfortable, I can't really say whether I think he's ready for e.g. calculus. But I will say that just because he is having some difficulty with these problems, it's not necessarily a sign that he [I]isn't[/I] ready.

It appears that your mental model for math education is that a student needs to show true mastery of the material at any given level before attempting to tackle a more advanced level. That sounds like a reasonable--maybe even an ideal--model to have. Unfortunately it doesn't particularly accord with the way that most people actually learn. The truth is that exposure to more advanced material can be one of the fastest ways to solidify understanding of the basic material. In my experience, both as a student and a teacher, it is a frequent occurrence that after being exposed to more advanced topics, the less advanced ones can be revisited with greater understanding; sometimes understanding that the student didn't even realize was missing.

And sure, true mastery of the basics would make learning the more advanced stuff easier. But it is far more common that students learn well by moving back and forth through the levels of difficulty, such that the new, more difficult topics help to reinforce the basics, and that reinforcement in turn helps make the difficult topics easier as they go. And that back-and-forth process ends up taking less total time than trying to achieve complete mastery of the basics and then moving on.

As an example, I would guess that a fair number of people in this forum have taken courses in both number theory and group theory, in that order. How many of those people discovered that many of the concepts in number theory just made more sense after learning group theory?

I think the exercises you are giving to Tomer are good ones for him, but IMHO the fact that he finds them difficult does not definitely mean that he isn't ready to learn the basics of calculus. Indeed, I would guess that he could learn the basics of calculus without too much trouble, and that after doing so he would find these problems to be easier.

[QUOTE=R.D. Silverman;218562]Throughout these exercizes let f(x) = a_n x^n + a_{n-1} x^(n-1) + .... a_0.

Suppose that its roots are x_0, x_1, .... x_n.
[/QUOTE]

Wait a sec, that doesn't sound right. :wink:

[quote=R.D. Silverman;218596]Huh? The first question is clear. You are given a polynomial and its
roots and asked to derive the polynomial whose roots are the reciprocals
of the given roots. Why isn't this understandable?????

The second question is somewhat related to the binomial theorem.[/quote]

For the first question,
Did you mean?:
For f(x)'s roots are x_0, x_1, ..., x_n, we have a polynomial with roots based on f(x)'s roots, which are 1/x_i.

[quote=R.D. Silverman;218598]But you keep claiming that you are ready for more advanced material!!
(calculus, college algebra, etc).

The polynomial exercizes are typical of ones that I encountered when taking
pre-calculus in high school. If you can not handle them, then you are
not, IMO, ready for more advanced material.

Would anyone else like to render an opinion?

The exercizes I gave should be typical for a high level honors high
school class.[/quote]

I did not claim such a thing (maybe three months ago, but not now).
And more then it, didn't claim I can't solve these...
Just, as you gave me Shanks' book (which was great) I wanted to make my solvements much more intelligent ( I mean, less "OUT OF IGNORANCE").
On my age (in school) things as Newton's binom, The remainder theorem, etc, are not taught (not in my country).
Why did you write "this is easy with calculus", I don't know anything in calculus.

[QUOTE=jyb;218629]Wait a sec, that doesn't sound right. :wink:[/QUOTE]

Oops. Wrong indices. One too many roots.......

[QUOTE=blob100;218632]For the first question,
Did you mean?:
For f(x)'s roots are x_0, x_1, ..., x_n, we have a polynomial with roots based on f(x)'s roots, which are 1/x_i.[/QUOTE]

Can someone else help? We seem to have a language problem.
I do not know how to express the problem any more clearly.

[quote=R.D. Silverman;218660]Can someone else help? We seem to have a language problem.
I do not know how to express the problem any more clearly.[/quote]
There is no language problem.
I meant:
The problem concerns a polynomial g(x) with roots of the form 1/x_i right?
When you denote x_i you mean: a root of f(x) (defined on the exercizes post before the first problem)?
Or, a given polynomial g(x) with roots of the form 1/n for any n (which is nothing interesting because every number is an inverse of another number).

[QUOTE=jyb;218627]
As an example, I would guess that a fair number of people in this forum have taken courses in both number theory and group theory, in that order. How many of those people discovered that many of the concepts in number theory just made more sense after learning group theory?
[/QUOTE]

I did it in reverse. I took algebra as an undergrad, but no number theory.
I did not learn any number theory until later.

[QUOTE]

I think the exercises you are giving to Tomer are good ones for him, but IMHO the fact that he finds them difficult does not definitely mean that he isn't ready to learn the basics of calculus. Indeed, I would guess that he could learn the basics of calculus without too much trouble, and that after doing so he would find these problems to be easier.[/QUOTE]

I mean no offense to Tomer, but from what I have seen, I can't
imagine that he has the ability right now to perform a epsilon-delta
proof for a limit problem. Even a simple one.

I talk with colleagues (University professors) all the time. Their biggest
complaint is that too many students do not have sufficient mastery
of high school level algebra/geometry/trig to tackle college calculus.
There are too many students taking too many bonehead "remedial"
courses. And they are even failing at those.

Have you read Andrei Toom's essay on the pseudo-education that is
taking place with math education today [in the US]? It is a terrific
essay.

[QUOTE=blob100;218661]There is no language problem.
I meant:
The problem concerns a polynomial g(x) with roots of the form 1/x_i right?
When you denote x_i you mean: a root of f(x) (defined on the exercizes post before the first problem)?
Or, a given polynomial g(x) with roots of the form 1/n for any n (which is nothing interesting because every number is an inverse of another number).[/QUOTE]

Huh? Where did you get this last bit of nonsense? I defined x_i.
Having defined x_i, 1/x_i also becomes well defined. Where are you pulling
'n' from?? What relation does it have to x_i???

I have commented before that you have a bad habit of introducing
extraneous variables without properly defining them.

We do have a language problem. x_i were [b]defined[/b] as the roots of a
certain polynomial. 1/x_i are therefore well defined.

Someone else needs to help.

[quote=R.D. Silverman;218665]
. x_i were [B]defined[/B] as the roots of a
certain polynomial. 1/x_i are therefore well defined.

.[/quote]

Ho, Great! This is what I wanted to hear.
f(x)=a_nx^n+a_(n-1)x^(n-1)+....+a_0.
g(x) have the roots 1/x_i.
g(x)=(a_na_0/a_o)x^n+(a_na_1/a_0)x^(n-1)+....+
(a_na_(n/2+1)/a_0)x^(n/2)+....+(a_na_n/a_0).
For even n.

[QUOTE=blob100;218667]Ho, Great! This is what I wanted to hear.
f(x)=a_nx^n+a_(n-1)x^(n-1)+....+a_0.
g(x) have the roots 1/x_i.
g(x)=(a_na_0/a_o)x^n+(a_na_1/a_0)x^(n-1)+....+
(a_na_(n/2+1)/a_0)x^(n/2)+....+(a_na_n/a_0).
For even n.[/QUOTE]

What is "a_na_0/a_o" etc. What is "a_na_n/a_0" ????? .......

And why for "even n"???

And the coefficients that you have written are not integers........

Note: The correct answer is quite simple.