Cubic with three real irrational roots.

In summary, the conversation discusses the possibility of expressing the roots of a cubic polynomial with integer coefficients as simple surd expressions. The argument is that if one root can be expressed in this form, then the other root must also be able to be expressed in the same form. However, the definition of "surd" is debated and it is concluded that not all cubic roots can be expressed as nested surds due to the possibility of complex roots. There is also a mention of Galois theory, which may provide more information on this topic.
  • #1
uart
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Considering the case of cubic polynomials with integer coefficients and three real but irrational roots. Is it true that it's impossible that all three roots can be in the form of simple surd expressions like [tex]r+s \sqrt{n}[/tex] (where r and s are rational and sqrt(n) is a surd). The argument is that if [tex]r+s \sqrt{n}[/tex] is a solution then you can show that [tex]r-s \sqrt{n}[/tex] must also be a solution.

Does anyone have any extra info or links to theorems or other useful info related to this.

Thanks.
 
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  • #2
It sounds like you have a reasonable proof of this. Do you know about Galois theory?
 
  • #3
No, it's not true if by "surd" you mean the usual definition: a root of a real number. Of course, any cubic equation can be solved by the cubic formula, giving the root as a combination of cube roots of an expression involving a square root. However, the result of the square may be imaginary and so the cube root is not a "surd".
 
  • #4
Hurkyl said:
It sounds like you have a reasonable proof of this. Do you know about Galois theory?

No I don't know about Galois theory but I just looked at a quick overview at http://en.wikipedia.org/wiki/Galois_theory.

No, it's not true if by "surd" you mean the usual definition: a root of a real number. Of course, any cubic equation can be solved by the cubic formula...

So would it be true to say that any cubic root could be expressed as a something like a nested surd then?
 
  • #5
uart said:
No I don't know about Galois theory but I just looked at a quick overview at http://en.wikipedia.org/wiki/Galois_theory.



So would it be true to say that any cubic root could be expressed as a something like a nested surd then?

I'm beginning to wonder if I know anything! My answer above was "no" if by surd you mean the "usual definition: a root of a real number". In order to give support for my contention, I "googled" on surd and found that MathWorld defines "surd" as "any irrational number"!
Wikpedia, however, defines "surd" as "an unfinished expression used in place of resolving a number's square, cube or other root, usually because the root is an irrational number.", my definition.

With the first of those definitions, "any cubic root could be expressed as a something like a nested surd" would be false because two of the cube roots might be complex rather than irrational numbers, the MathWorld definition of "surd".

With the second of those definitions, "any cubic root could be expressed as a something like a nested surd" is still false because the nested roots might be roots of complex numbers rather than real numbers, the Wikpedia definition of "surd".

So my answer to your question is still NO, even though the roots of every cubic equation can be written as combinations of nested second and third roots, those roots or the numbers in the roots might be complex rather than real and, thus, not surds!
 
  • #6
Yeah that's interesting Halls. So Mathworld just considers the term "Surd" to be an antiquated term for an irrational. Note however that they do refer to the term "Quadratic Surd" as having the meaning that I (and probably yourself) normally think of as a Surd. http://mathworld.wolfram.com/QuadraticSurd.html
 

FAQ: Cubic with three real irrational roots.

1. What is a cubic with three real irrational roots?

A cubic with three real irrational roots is a type of polynomial function with a degree of three, meaning the highest exponent on the variable is three. It has a total of three roots, or solutions, that are all real numbers and cannot be expressed as a ratio of two integers.

2. How can you determine if a cubic has three real irrational roots?

A cubic with three real irrational roots can be determined by using the discriminant, which is found by taking the square root of the expression b^2-4ac, where a, b, and c are the coefficients of the cubic function. If the discriminant is positive and not a perfect square, then the cubic has three real irrational roots.

3. Can a cubic with three real irrational roots have any rational roots?

Yes, a cubic with three real irrational roots can have rational roots as well. The term "irrational" only refers to the roots that cannot be expressed as a ratio of two integers. A cubic can have a combination of rational and irrational roots.

4. How are cubic functions with three real irrational roots graphed?

Cubic functions with three real irrational roots can be graphed by first finding the x-intercepts, which are the points where the function crosses the x-axis. These points are the three real irrational roots of the cubic. The graph will have a shape of a "W" or an "M" depending on the sign of the leading coefficient.

5. Are cubic functions with three real irrational roots common in real-world applications?

Cubic functions with three real irrational roots are not as common as other types of functions, such as linear or quadratic functions. However, they can still be seen in real-world applications, especially in physics and engineering, where they are used to model various phenomena such as projectile motion and fluid dynamics.

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