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

[Kate2010]

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.

 

[Mark44]

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.

 

[VeeEight]

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.