Help with a polynomial question?

[latesummerrain]

Prove that x^3 -3(α^2)(x^2) +3(α^3)x + α^4 =0 has exactly one real root.

 

[JeffM]

Prove that x^3 -3(α^2)(x^2) +3(α^3)x + α^4 =0 has exactly one real root.

I presume that a is defined as a real number. Is that presumption correct?

I have not worked out a proof, but one possible route to take is to start from the fundamental theorem of algebra. From there, I might try a proof by contradiction. I do not promise that will succeed, but it seems to me to be a straight forward approach. Perhaps as a preliminary you may want to show that if a = 0, the proposition is trivial.

Why don't you give that approach a try? If it seems to lead nowhere, show us what you have done, and maybe we can see a way to proceed along that line. If not, someone can suggest another idea.

EDIT: I worked a bit on this approach. It may work, but, if so, I do not think it will be a very concise proof. Consequently, it may not be a good approach. Of course, one problem about helping with proofs is that we have no idea what theorems were previously established that you can use.