Are there no good ways to solve polynomial equations?

[Trurl]

How do we solve polynomial equations? I only know how to simplify them. If you could solve this polynomial say N = 85 it would earn a million dollars. Remember who gave it to you.

 

x^3 = N^2 * (x^2/(N^2/x + x))

 

Solve for x

[uncool]

1) That is not a polynomial equation; note the division.

2) Basic manipulation can turn it into the equation x^4 = 0, which means that it has no solutions (since when x = 0, the RHS involves division by 0).

3) What, precisely, do you mean by "solve"? Numerically? Using radicals? Finding a minimal polynomial?

[Trurl]

 I want to solve for x. Yes I know that I did not isolate x, but I know the equation has a solution. I cannot get Mathematica to solve it.

I was wondering if there is anyway to solve such equations. It is a recurring theme. It is not like a text book problem that is designed to be solved. I keep making complex equations I can’t solve.

I don’t know of any way. The rules I have learned aren’t sufficient to solve. The equation is simple enough,, but I thought it would have an easy solution. Does this equation mean anything to you?

[Trurl]

N = x*y

 

Let N = 85

 

(N^2 + x^3) / N = N + (x^2 / (N^2/ x + x) * N

 

Y1graph = (N^2 + x3) / N

 

Y2graph =  N + (x^2 / (N^2/ x + x) * N

 

Y1graph - Y2graph < 1

 

I can graph them where it intersects. I can also equate the inequality,but I want to solve for x knowing only N. Not plugging in N and x.

 

 

 

I was thinking I could use the derivative of each side. Until x^3 = 6x in the first equation I gave you in my first post. This would not work with the ygraph because their graphs are only equal when N = 85.

 

 

I don’t have any clue how to solve for x. The rules of simplification I know have failed me.

[HallsofIvy]

On 11/9/2020 at 9:36 PM, Trurl said:

How do we solve polynomial equations? I only know how to simplify them. If you could solve this polynomial say N = 85 it would earn a million dollars. Remember who gave it to you.

 

x^3 = N^2 * (x^2/(N^2/x + x))

 

Solve for x

First, as you have already been told this is NOT a "polynomial equation".  

It can be written x^3= (N^2xc{N^2/x+ x)$.

Multiply both sides by that denominator, N^2/x+x.

N^2x^2+ x^4= N^2x^2.

And that reduces to x^4= 0.  Can you solve THAT polynomial equation?

Edited November 15, 2020 by HallsofIvy

[Trurl]

And that reduces to x^4= 0.  Can you solve THAT polynomial equation?

 

 

That is exactly what I am saying: the traditional simplification rules do not solve the equations accurately.

 

I’m saying there must be new rules needing discovered that would solve solutions that were once impossible. Like calculus was discovered as a need to solve physics.

 

Graph Y1graph and Y2graph and see the solution of pnp. We have more options with a plot. We can see where they intersect. I know the equations don’t seem to be of any value, but a graph opens up new approaches.

 

Is a computer plot in polynomial time? I don’t know if it is simpler then recursion, but it seems fast.

 

When I was in college we drew a shear diagram and graph it using graphical calculus to find the moment diagram. Pretty cool finding the area does what is computational hard.

 

I do have graphs and code I’d like to share.

[HallsofIvy]

I have no idea what you mean by "traditional simplification rules".  

I said "And that reduces to x^4= 0.  Can you solve THAT polynomial equation?"

Your response was "That is exactly what I am saying: the traditional simplification rules do not solve the equations accurately."

??  That was a "traditional simplification rule" that did solve the equation.  What I would mean by "traditional simplification rules" (again, I don't know what you mean by that) solve those equations that can be solved in terms of roots, such as this one, very nicely.   Of course, it has been known for a long time that there were polynomial equations that cannot be solved in terms of roots.  There are other methods that work, to get at least numerical approximations, to any accuracy, for those

[studiot]

16 hours ago, Trurl said:

I’m saying there must be new rules needing discovered that would solve solutions that were once impossible. Like calculus was discovered as a need to solve physics.

 

When I was in college we drew a shear diagram and graph it using graphical calculus to find the moment diagram. Pretty cool finding the area does what is computational hard.

 

You ae right that there are graphical methods that can solve polynomials, and other equations, to any required degree of accuracy.

For polynomials Lills (1867) method is one such.

But please get your definition of a polynomial expression.

It is of the form

a_nXⁿ + a_(n-1)X^(n-1)+ a_(n-2)X^(n-2) ...........a₀X + b                        Where the a_n, b are coefficients , not all zero

Conventionally you would generate an equation by setting this polynomial expression equal to zero.

Lills method was originally in French,

An English version appears in Theory of Equations, Turnbull, 1939 along with other methods of dealing with polynomials eg that of Horner's reduction of of degree.

Lills method is also dealt with at great length by

Cremona Graphical Calculus and Reciprocal figures. This also deals with a whole raft of graphical methods.
 

The original 1888 translation book was in Italian but has been translated to English in 1977 by Beare.

Finally Ewart (1919) Elements of Graphic Dynamics

also  deals with this and other useful subjects such as graphical integration and graphical differentiation.

[Trurl]

Here are 2 equations that are not algebraically useful, but the graph is gold dust.

 

http://www.constructorscorner.net/Files/GraphicCalcSimple.pdf

 

 

GraphicCalcSimple.pdf