Prove that there exists an x such that x[SUP]3[/SUP] = 2

  • Thread starter Kate2010
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In summary, the problem is to prove the existence of an x such that x3 = 2. The previous work done using the intermediate value theorem shows that every monic polynomial of odd degree has a real root. Therefore, considering x3 - 2 = 0 as a monic polynomial of odd degree, it is known that it has a real root. This can be taken as the x that is being sought for, although it may not fully prove its existence.
  • #1
Kate2010
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Homework Statement



Prove that there exists an x such that x3 = 2

Homework Equations





The Attempt at a Solution



I have deduced in an earlier part of the question, using the intermediate value theorem, that every monic polynomial of odd degree has a real root.

So if I consider x3 - 2 = 0, as a monic polynomial of odd degree, I know that it has a real root. Can I just say that this is the x that I am looking for? I don't feel like I've really proved it fully.
 
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  • #2
Kate2010 said:

Homework Statement



Prove that there exists an x such that x3 = 2

Homework Equations





The Attempt at a Solution



I have deduced in an earlier part of the question, using the intermediate value theorem, that every monic polynomial of odd degree has a real root.

So if I consider x3 - 2 = 0, as a monic polynomial of odd degree, I know that it has a real root. Can I just say that this is the x that I am looking for? I don't feel like I've really proved it fully.
Based on your problem statement, all you need to show is existence, and the previous work you did shows that for this polynomial.
 
  • #3
If all you have to do is prove the existence of such an x then why don't you simply find x, which isn't very difficult. Unless there is something I'm missing.
 

1. Can you explain the statement "there exists an x such that x3 = 2" in simpler terms?

The statement means that there is a number (x) that when cubed (multiplied by itself three times), the result is 2.

2. Why is it important to prove that such a number exists?

This proof is important in mathematics because it helps us understand the concept of irrational numbers and their existence. It also allows us to solve equations involving cubic roots.

3. How do scientists and mathematicians approach proving the existence of a number in this context?

They use a method called contradiction, where they assume that the statement is false and then use logical reasoning and mathematical operations to prove that it leads to a contradiction. This contradiction then proves that the original statement is true.

4. Is there only one number that satisfies the equation x3 = 2?

No, there are actually three numbers that satisfy this equation: the positive real number 2^(1/3), the negative real number -2^(1/3), and the imaginary number i*2^(1/3).

5. Can you provide an example of how this statement can be proven?

One way to prove this statement is by using the intermediate value theorem, which states that if a continuous function (in this case, x^3) has opposite signs at two points (one above and one below the x-axis), then there must be at least one root (x value where the function equals 0) between those two points. In this case, we can show that the function x^3 - 2 has opposite signs at x=1 and x=2, thus there must be a root between those two points, which proves that there exists an x such that x^3 = 2.

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