[quote=R.D. Silverman;218687]What is "a_na_[B]0[/B]/a_[B]o[/B]" 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.[/quote]
a_na_n/a_0 is the product of a_n and itself (a_n)^2 devided by a_0
(you defined these by yourself).
"a_na_[B]0[/B]/a_[B]o[/B]"--> a_na_0/a_0.
These does not need to be integers.

For f(x)=a.nx^n+a.(n-1)x^(n-1)+....+a.0
For x^k the coefficient is ((a.n * a.P)/ a.0) where P means the "paralleled" for k, defined: if n is the order of the polynomial, |n-k|=P-0=P.
Where k is any of the powers of x in the polynomial (1,2...,n).
Example:
(x-1)(x+2)(x-2)=f(x)
=x^3-x^2-4x+4
(x-1)(x+1/2)(x-1/2)=g(x)
=x^3-x^2-(1/4)X+1/4
The paralleled for 3 is 0, 2-->1, etc.

The second problem:
Again using P as a "paralleled" of k, x^k's coefficient is (where the order of the polynomial is N):
-/+(The sum of the product combinations between the roots, where every factor in the sum is a product of P roots),
For order 3, the coefficient of x^2 is -(x3+x2+x1).
The value is minus or plus depends on N-k, If it is odd, minus, if it is even, plus.

[QUOTE=R.D. Silverman;218663]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.[/QUOTE]

I agree that he probably doesn't have that ability right now. But no one is born knowing how to perform epsilon-delta proofs. It has to be taught--and learned. And it's hard to say based on what we've seen, but I would guess that it's something he would be able to learn if given good instruction. And perhaps you're thinking that his track record of learning things doesn't look too good in this forum. But let us agree that this forum is a [I]lousy[/I] medium for learning mathematics.

[QUOTE=R.D. Silverman;218663]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]

Yes, I agree with the vast majority of what Toom has to say. (Some of his specific examples, like having to snatch calculators out of the hands of students so they will actually [I]think[/I], hit very close to home. I have had some of those exact experiences.) You won't find me defending the vague state of primary school mathematics education in this country. And indeed, Toom even makes a point about students trying to learn material which is simply too advanced for their abilities, which is pretty much what you're talking about.

Look, what I said above can certainly be taken too far. I don't advocate trying to force advanced material on people with minimal mathematical background (even if they're requesting it). It's just that I see time and time again, in mathematics and in other disciplines, this phenomenon where exposure to advanced topics, methods and mindsets can really help understanding of the more basic topics, even when the student thought they already understood the basic topics. So there's some level of problem difficulty beyond which a failure to solve the problem is not necessarily a barrier to learning more advanced topics. That difficulty level is probably different for every student, but I don't think it's infinitely high, as you seem to.

[QUOTE=blob100;218706]a_na_n/a_0 is the product of a_n and itself (a_n)^2 devided by a_0
(you defined these by yourself).
"a_na_[B]0[/B]/a_[B]o[/B]"--> a_na_0/a_0.
These does not need to be integers.[/QUOTE]

But your last expression a_na_0/a_0 is just a_n........

Your answer is not correct. This problem is quite simple, only takes
two steps, and yields a very simple answer that does not involve
ratios of coefficients. Indeed, the derivation does not even involve
arithmetic operations performed on the coefficients.......

[QUOTE=blob100;218716]The second problem:
Again using P as a "paralleled" of k, x^k's coefficient is (where the order of the polynomial is N):
-/+(The sum of the product combinations between the roots, where every factor in the sum is a product of P roots),
For order 3, the coefficient of x^2 is -(x3+x2+x1).
The value is minus or plus depends on N-k, If it is odd, minus, if it is even, plus.[/QUOTE]

Without quibbling about the wording, this is essentially correct.
The coefficient is (-1)^k times the sum of all products of the coefficients taken k at a time.

[quote=R.D. Silverman;218732]But your last expression a_na_0/a_0 is just a_n........

Your answer is not correct. This problem is quite simple, only takes
two steps, and yields a very simple answer that does not involve
ratios of coefficients. Indeed, the derivation does not even involve
arithmetic operations performed on the coefficients.......[/quote]

a_na_0/a_0 is just a_n........ true, but I wanted to show the structure, not the best and most comfortable expression.

[QUOTE=jyb;218731]I agree that he probably doesn't have that ability right now. But no one is born knowing how to perform epsilon-delta proofs. It has to be taught--and learned. And it's hard to say based on what we've seen, but I would guess that it's something he would be able to learn if given good instruction. And perhaps you're thinking that his track record of learning things doesn't look too good in this forum. But let us agree that this forum is a [I]lousy[/I] medium for learning mathematics.
[/QUOTE]

Agreed. But I wasn't thinking so much about the aspect of learning how
to do epsilon-delta proofs specificallty. Instead, based upon what I have
seen, he has, so far, been unable to put together a rigorous derivation of
anything. He fails to heed my advice about defining variables. He fails
to heed my advice about introducing superfluous variables. (which his posts
have shown only confuses things). He ignores my advice when I give hints.
I have suggested that he try to present a semi-formal proof where he gives a
reason for each step based upon known results and previously derived
results. He doesn't seem to be able to do this either. He is very very
sloppy. He resorts too much to "handwaving".

Epsilon-delta proofs are usually the first instance where students must
show something that depends on more than one quantifier. One must be
very organized to put together such proofs. I don't think he is ready.

He doesn't seem able to handle the first polynomial algebra exercize that
I gave, and this one is totally trivial.

For the first question:
As you said, there is a missunderstanding about the problem.
So, I will give a private case (n=3) to show what I have understood:
f(x)=(x+a)(x+b)(x+c)= x^3+(a+b+c)x^2+(ca+cb+ab)x+abc (By simple algebra).
The roots are -a,-b,-c.
In your problem asks to find the coefficients of the polynomial g(x) where it's roots are the inverses of f(x)'s roots.
g(x)=(x+1/a)(x+1/b)(x+1/c)=
x^2+((ca+cb+ab)/abc)x^2+((a+b+c)/abc)x+1/abc

For f(x)=(a.n)x^n+a.(n-1)x^(n-1)....a.0.
a.i denotes the coefficient (integers) of x^i, n integer (the order of the polynom).
Do you remember how I defined P on the last posts?
The proposition:
The coefficient of x^k in the polynom g(x) (the polynom with the inverse roots of f(x)) are:
(a.P*a.n)/(a.0).
For k integer<n, P the paralleled of k.

[QUOTE=blob100;218855]For the first question:
As you said, there is a missunderstandment about the problem.
So, I will give a private case shows what I understand you want me to do in the problem.
f(x)=(x+a)(x+b)(x+c)
(x^2+(a+b)x+ab)(x+c)=x^3+cx^2+(a+b)x^2+c(a+b)x+abx+abc=
x^3+(a+b+c)x^2+(ca+cb+ab)x+abc.
The roots are -a,-b,-c.
Your problem asks to find the coefficients of the polynomial g(x) where it's roots are the inverses of f(x)'s roots.
g(x)=(x+1/a)(x+1/b)(x+1/c)=
x^2+((ca+cb+ab)/abc)x^2+((a+b+c)/abc)x+1/abc
[/QUOTE]

Sorry, but this isn't even close.

One can express the coefficients of g(x) very simply in terms of the coefficients of f(x). You are making the problem much more difficult than
it is.

And once again, you are introducing extraneous variables. The entire
problem can be solved using ONLY the variables presented in the problem
itself. You are also trying to solve the general case by presenting a
specific example . Don't.



[QUOTE]

For f(x)=a.nx^n+a.(n-1)x^(n-1)....a.0.
a.n integers, n integer (the order of the polynom).
Do you remember how I defined P on the last posts?
The proposition:
The coefficient of x^k in the polynom g(x) (the polynom with the inverse roots of f(x)) are:
(a.P*a.n)/(a.0).
For k integer<n, P the paralleled of k.[/QUOTE]

This is irrelevant to the problem.

[QUOTE=blob100;218855]The coefficient of x^k in the polynom g(x) (the polynom with the inverse roots of f(x)) are:
(a.P*a.n)/(a.0).
For k integer<n, P the paralleled of k.[/QUOTE]

I think it's standard form to have integer coefficients.