[quote=R.D. Silverman;218558]This works nicely.[/quote]

Thanks.

[QUOTE=blob100;218552]

<snip>
.[/QUOTE]

How about some exercizes on polynomial algebra?

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.
a_i are integers.

(1) Find the coefficients of a polynomial whose roots are 1/x_0, 1/x_1, ... 1/x_n.

Is this polynomial unique?

(2) Express the coefficient of x^k as a function of the roots.

(3) Prove the remainder theorem:

The remainder left when f(x) is divided by (x-r) equals f(r).

(4) Prove the rational roots theorem:

If alpha = a/b is a rational root of f(x) then b divides a_n
and a divides a_0.

(5) Prove that any polynomial of odd degree must have a real root.

This is easy with Calculus. Prove it without the Intermediate Value Thm.


(6) Prove that if a_n = 1, (known as a monic polynomial), then
any rational root must be an integer.

(7) Solve x^4 + x^3 + x^2 + x + 1 = 0. Hint: use the result from problem
(1).

Note that this problem can be solved without knowing how to solve quartics
in general.

(8) Suppose all of the coefficients of f(x) equal 1. Prove that the absolute
value of all of the roots must equal 1.

[quote=R.D. Silverman;218562]How about some exercizes on polynomial algebra?

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.
a_i are integers.

(1) Find the coefficients of a polynomial whose roots are 1/x_0, 1/x_1, ... 1/x_n.

Is this polynomial unique?

(2) Express the coefficient of x^k as a function of the roots.

(3) Prove the remainder theorem:

The remainder left when f(x) is divided by (x-r) equals f(r).

(4) Prove the rational roots theorem:

If alpha = a/b is a rational root of f(x) then b divides a_n
and a divides a_0.

(5) Prove that any polynomial of odd degree must have a real root.

This is easy with Calculus. Prove it without the Intermediate Value Thm.


(6) Prove that if a_n = 1, (known as a monic polynomial), then
any rational root must be an integer.

(7) Solve x^4 + x^3 + x^2 + x + 1 = 0. Hint: use the result from problem
(1).

Note that this problem can be solved without knowing how to solve quartics
in general.

(8) Suppose all of the coefficients of f(x) equal 1. Prove that the absolute
value of all of the roots must equal 1.[/quote]
As you said, my skills are quiet losy in this area, will you please give me a name of a book I can read on the topic?

[QUOTE=blob100;218565]As you said, my skills are quiet losy in this area, will you please give me a name of a book I can read on the topic?[/QUOTE]

This is basic pre-calculus algebra. You should have seen this stuff in
2nd/3rd year secondary school algebra.

The Schaum Outline Series should cover this. I can also recommend a
superb high-school level textbook: Introductory Modern Analysis
that is surely long out of print. This is the text I used.

[QUOTE=R.D. Silverman;218571]This is basic pre-calculus algebra. You should have seen this stuff in
2nd/3rd year secondary school algebra.

The Schaum Outline Series should cover this. I can also recommend a
superb high-school level textbook: Introductory Modern Analysis
that is surely long out of print. This is the text I used.[/QUOTE]

Here it is:

[url]http://www.amazon.com/Modern-Introductory-Analysis-Mary-Dolciani/dp/0395350484[/url]

[quote=R.D. Silverman;218571]This is basic pre-calculus algebra. You should have seen this stuff in
2nd/3rd year secondary school algebra.

The Schaum Outline Series should cover this. I can also recommend a
superb high-school level textbook: Introductory Modern Analysis
that is surely long out of print. This is the text I used.[/quote]

I have seen these stuff on these years of secondary school, but it does not say these exersizes are on the same level with the secondary school's level (there are many levels for everything, you can teach a theory on a verry deep way and on the same time the opposite).

I am giong to get "Undergraduate Algebra" by Serge Lang in few weeks and I took from TAU's library "A course of pure mathematics" by G.H hardy,
Are these discussing the area of polynomials as I needed for these exercises?

[quote=R.D. Silverman;218562]How about some exercizes on polynomial algebra?

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.
a_i are integers.

(1) Find the coefficients of a polynomial whose roots are 1/x_0, 1/x_1, ... 1/x_n.

Is this polynomial unique?

(2) Express the coefficient of x^k as a function of the roots.

.[/quote]

The first question isn't understandable.
The roots 1/x_0, 1/x_1, ... can be shown as roots of the form x_i instead of 1/x_i (every number have an inverse).

The second question is concerning the famous Newton's Binom.

[quote=R.D. Silverman;218571]This is basic pre-calculus algebra. You should have seen this stuff in
2nd/3rd year secondary school algebra.

The Schaum Outline Series should cover this. I can also recommend a
superb high-school level textbook: Introductory Modern Analysis
that is surely long out of print. This is the text I used.[/quote]

I don't have the whole Schaum Outline series, I have:
Advanced Infinitesimal arithmetic and complex variables.

[quote=blob100;218575]I have seen these stuff on these years of secondary school, but it does not say these exersizes are on the same level with the secondary school's level (there are many levels for everything, you can teach a theory on a verry deep way and on the same time the opposite).

I am giong to get "Undergraduate Algebra" by Serge Lang in few weeks and I took from TAU's library "A course of pure mathematics" by G.H hardy,
Are these discussing the area of polynomials as I needed for these exercises?[/quote]


Stay away from Lang's books!! They are terrific [B]references[/B] if
you know the material, but make horrible textbooks. Also, this book
is about modern algebra: boolean algebras, monoids, groups, rings, fields.
It will assume that you [b]already[/b] know the polynomial results that are given by my exercizes.

I mean no offense, but you keep trying to run before you can walk.
Master the secondary school stuff first, then move on.

[QUOTE=blob100;218576]The first question isn't understandable.
The roots 1/x_0, 1/x_1, ... can be shown as roots of the form x_i instead of 1/x_i (every number have an inverse).

The second question is concerning the famous Newton's Binom.[/QUOTE]

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=blob100;218575]I have seen these stuff on these years of secondary school, but it does not say these exersizes are on the same level with the secondary school's level (there are many levels for everything, you can teach a theory on a verry deep way and on the same time the opposite).
[/QUOTE]

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.