Math Is Fun Forum

God

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Registered: 2005-08-25

Posts: 59

A cubic...

Can the cubic 0 = x^3 - x - 1 be solved by means of simple algebra (without the cubic formula)?

Jai Ganesh

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Registered: 2005-06-28

Posts: 48,385

Re: A cubic...

x³ - x - 1 = 0
x³ - x = 1
x(x²-1)=1
x = 1/(x²-1)
Does this cubic equation have a real solution?

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Ricky

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Registered: 2005-12-04

Posts: 3,791

Re: A cubic...

No, it can't be factored.  The only real factor is a real number x~1.32.  So any attempts to factor it would be futile.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: A cubic...

It has one real root, but I used the cubic formula:
x≈1.32471795724474602596090885448...

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Ricky

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Registered: 2005-12-04

Posts: 3,791

Re: A cubic...

So what is the exact root of it, krassi?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

krassi_holmz

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Registered: 2005-12-02

Posts: 1,905

Re: A cubic...

No. Do you REALLY want it?

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Ricky

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Registered: 2005-12-04

Posts: 3,791

Re: A cubic...

If you use the cubic formula, shouldn't you come up with the exact root?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,385

Re: A cubic...

I did not know the roots of a cubic equation and looked for it on the net. I got to this pdf. Ricky, you can find the root of the incomplete cubic equation using the formulae.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: A cubic...

The roots of quadratic equation ax^2+bx+c:

Last edited by krassi_holmz (2006-01-04 22:08:31)

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krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: A cubic...

Cubic equation:

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krassi_holmz

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Registered: 2005-12-02

Posts: 1,905

Re: A cubic...

And quadric equation (useless):

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krassi_holmz

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Registered: 2005-12-02

Posts: 1,905

Re: A cubic...

For you question we get:

Last edited by krassi_holmz (2006-01-04 22:12:44)

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krassi_holmz

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Registered: 2005-12-02

Posts: 1,905

Re: A cubic...

Equations >4 are unsolvable exactin radicals. (Galoa)

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krisper

Member

Registered: 2005-12-20

Posts: 19

Re: A cubic...

I am sure you could find this without using the cubic formula.
x^3 - x - 1 = 0
x(x^2-1) = 1
if x = 0, the equasion has no meaning, so x must differ from 0 (x<>0); Now we can devide by x.
x^2 - 1 = 1/x.
We draw the graphics of these two expressions - x^2 - 1 and 1/x. After that we check where they cross eachother. This happens only in I quadrant which means that x^3 - x - 1 = 0 has only one real root. Afterwards doing some calculas we find that the root is somewhere between 1 and √2. Now we have to use the tangents and make some calculations and I am sure we will get the exact value of this root.

Last edited by krisper (2006-01-04 22:16:42)

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krassi_holmz

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Registered: 2005-12-02

Posts: 1,905

Re: A cubic...

this is the exact root!

For you question we get:

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Jai Ganesh

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Registered: 2005-06-28

Posts: 48,385

Re: A cubic...

Very good posts and images uploaded, krassi_holmz. Those would be very useful. BTW, did you know the insolubility of the quintic equation was shown first by Neils Henrik Abel, the Norwegain Mathematician?

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: A cubic...

Yes, I've read very much about Abel.

And the Galous theory simplifies the result.
The solubility of an equation depends of the structude if its Galous group.

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krassi_holmz

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Registered: 2005-12-02

Posts: 1,905

Re: A cubic...

The general quintic can be solved in terms of Jacobi theta functions, as was first done by Hermite  in 1858. Kronecker  subsequently obtained the same solution more simply, and Brioschi also derived the equation.

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God

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Registered: 2005-08-25

Posts: 59

Re: A cubic...

Lol.thanks... wow... that looks so complicated lol

krassi_holmz

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Re: A cubic...

That's why I prefer the approximated result.

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God

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Registered: 2005-08-25

Posts: 59

Re: A cubic...

I guess you might as well just use Newton's recursion for these things...

Last edited by God (2006-01-08 09:41:58)