Math Is Fun Forum

CoRRupt3d

Guest

Equation without Integer Roots

My friend asked me to solve for the roots of

. Supposedly, the roots are

and

.

My question: Where do you even begin when given a polynomial such as in this form and how would you solve it?

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Equation without Integer Roots

Hm... I don't think those are the roots of

EDIT: But the roots of

are supposedly

and

.

Last edited by evene (2016-06-16 07:55:01)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Equation without Integer Roots

You are correct, those are the roots.

I would do it numerically, deflating out as I go. When I have them all I would apply the PSLQ to get the form you desire.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Equation without Integer Roots

...

I substituted

to get

We want the roots, so setting

will give us the roots.

Last edited by evene (2016-06-16 08:25:30)

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Equation without Integer Roots

Full work:

Let

. From that, we have

which simplifies into

.

Setting

and we get

. Equating terms together, we get

and

. Since

, we have

. So

for

.

We want the real part so we use the formula

to get the roots

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Equation without Integer Roots

First check the geometry of the roots.

Make use of Newton's iteration because it has quadratic convergence and the roots are well spaced apart.

x0 = 2 yields xn = 1.77091205130642

x0 = 1 yields xn = 1.136129493462312

x0 = 0 yields xn = .2410733605106461

x0 = -1 yields xn = -0.7092097740850714

x0 = -1.5 yields xn = -1.497021496342203

x0 = -2 yields xn = -1.941883634852104

Then take them over to the ISC or PSLQ them yourself to get the exact analytical answers.

.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Equation without Integer Roots

Okay, that works too!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Equation without Integer Roots

Hi;

There must be a couple of more ways. This polynomial looks like it has been economized...

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Equation without Integer Roots

I feel like my follow through is already simple enough. You simply use All Roots of Unity to solve the problem...

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Equation without Integer Roots

Wait, Bobbym, can you use your ISC thing to compute the roots of

? Setting

does not work.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Equation without Integer Roots

Hi;

That is the problem with methods like that. The numerical technique is more robust and can deal with a much wider range of problems but you must be able to program a bit. Also, and this changing you were not able to get nice clean closed forms. With the arrival of PSLQ that is no longer the case. Mind you, there is much work to be done along these lines as the technique is still in its infancy.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.