Polynomials

[Jon]

Polynomial Roots to any Degree
Vector Calculus Method

The projection of the curve T onto the normal N results in a logarithmic
solution. However, the Taylor coefficients for the logarithmic
contribution out-differentiate the polynomial, so it is subtracted as
error from the final solution.


http://207.69.235.90/members/jon8338/polynomial/id7.html

Jon Giffen

[Gregory Toomey]

Post by Jon
Polynomial Roots to any Degree
Vector Calculus Method

We has enough of this nonsense last month.

http://mathworld.wolfram.com/QuinticEquation.html
Unlike quadratic, cubic, and quartic polynomials, the general quintic cannot
be solved algebraically in terms of a finite number of additions,
subtractions, multiplications, divisions, and root extractions, as
rigorously demonstrated by Abel (Abel's impossibility theorem) and Galois.


gtoomey

[Proginoskes]

Post by Gregory Toomey

Post by Jon
Polynomial Roots to any Degree
Vector Calculus Method

We has enough of this nonsense last month.
http://mathworld.wolfram.com/QuinticEquation.html
Unlike quadratic, cubic, and quartic polynomials, the general quintic cannot
be solved algebraically in terms of a finite number of additions,
subtractions, multiplications, divisions, and root extractions, as
rigorously demonstrated by Abel (Abel's impossibility theorem) and Galois.

That argument doesn't work, since he's now thrown in the natural log
function ...
-- Christopher Heckman

[Gregory Toomey]

Post by Proginoskes

Post by Gregory Toomey

Post by Jon
Polynomial Roots to any Degree
Vector Calculus Method

We has enough of this nonsense last month.
http://mathworld.wolfram.com/QuinticEquation.html
Unlike quadratic, cubic, and quartic polynomials, the general quintic
cannot be solved algebraically in terms of a finite number of additions,
subtractions, multiplications, divisions, and root extractions, as
rigorously demonstrated by Abel (Abel's impossibility theorem) and Galois.

That argument doesn't work, since he's now thrown in the natural log
function ...
-- Christopher Heckman

Its not an argument like you hear at the pub, its a proof thats been around
since the 1800s.

You can throw in logarithms as well but it makes no difference.

gtoomey

[Proginoskes]

Post by Gregory Toomey

Post by Proginoskes

Post by Gregory Toomey

Post by Jon
Polynomial Roots to any Degree
Vector Calculus Method

We has enough of this nonsense last month.
http://mathworld.wolfram.com/QuinticEquation.html
Unlike quadratic, cubic, and quartic polynomials, the general quintic
cannot be solved algebraically in terms of a finite number of additions,
subtractions, multiplications, divisions, and root extractions, as
rigorously demonstrated by Abel (Abel's impossibility theorem) and Galois.

That argument doesn't work, since he's now thrown in the natural log
function ...
-- Christopher Heckman

Its not an argument like you hear at the pub, its a proof thats been around
since the 1800s.
You can throw in logarithms as well but it makes no difference.

I hadn't heard that (about Abel's proof also working with logarithms). What
else can you add and still be unable to find the roots of general
polynomials?
-- Christopher Heckman

[Virgil]

Post by Proginoskes

Post by Gregory Toomey

Post by Proginoskes

Post by Gregory Toomey

Post by Jon
Polynomial Roots to any Degree
Vector Calculus Method

We has enough of this nonsense last month.
http://mathworld.wolfram.com/QuinticEquation.html
Unlike quadratic, cubic, and quartic polynomials, the general quintic
cannot be solved algebraically in terms of a finite number of additions,
subtractions, multiplications, divisions, and root extractions, as
rigorously demonstrated by Abel (Abel's impossibility theorem) and Galois.

That argument doesn't work, since he's now thrown in the natural log
function ...
-- Christopher Heckman

Its not an argument like you hear at the pub, its a proof thats been around
since the 1800s.
You can throw in logarithms as well but it makes no difference.

I hadn't heard that (about Abel's proof also working with logarithms). What
else can you add and still be unable to find the roots of general
polynomials?
-- Christopher Heckman

Just about anything short of elliptic functions will leave some general
polynomial functions unsolvable, but I don't think that Abel's or
Galois' proofs cover anything beyond a finite number of additions,
subtractions, multiplications, divisions, and root extractions.

[Brian Smith]

Post by Jon
Polynomial Roots to any Degree
Vector Calculus Method
The projection of the curve T onto the normal N results in a logarithmic
solution. However, the Taylor coefficients for the logarithmic
contribution out-differentiate the polynomial, so it is subtracted as
error from the final solution.

Jon, there is a circular argument in your method.

When you construct the equation '-a_0 = Integ (N*dT) dt', the equation
reduces back to the original polynomial. That circular argument means
that your method fails.

Also, I would like your to see your application of your method to the
equation x^3 - ax^2 + X - a = 0, with 'a' as a variable. If you
cannot extract 'a' as a root, then your method is useless to get any
exact values.

Post by Jon
http://207.69.235.90/members/jon8338/polynomial/id7.html
Jon Giffen

[Proginoskes]

Post by Brian Smith

Post by Jon
Polynomial Roots to any Degree
Vector Calculus Method
The projection of the curve T onto the normal N results in a logarithmic
solution. However, the Taylor coefficients for the logarithmic
contribution out-differentiate the polynomial, so it is subtracted as
error from the final solution.

Jon, there is a circular argument in your method.
When you construct the equation '-a_0 = Integ (N*dT) dt', the equation
reduces back to the original polynomial. That circular argument means
that your method fails.

This part of his "method" isn't incorrect, just not stated as is normally
done. He's saying that if (his first formula) holds, then (his second
formula, with the integral) also holds.

Post by Brian Smith
Also, I would like your to see your application of your method to the
equation x^3 - ax^2 + X - a = 0, with 'a' as a variable. If you
cannot extract 'a' as a root, then your method is useless to get any
exact values.

He can't even solve a cubic (which is what his example reduces to). He
notes that 1 is a root of t^4 + t^3 + t^2 + t - 4, and thus wants to
find the roots of t^3 + 2 t^2 + 3 t + 4, which he can evidently only
approximate. A message to Jon: If you claim your procedure works, at
least find a polynomial where the procedure actually gives you the
right answers, and approximations don't count. (You've been told this
numerous times!)
-- Christopher Heckman