Question: Prove that the equation has only one real root.
The existence of at least one real root follows from the fact that the above polynomial is an odd power.
Let us prove the uniqueness of such a root by contradiction.
Suppose there exist two roots . Then, in the interval the given polynomial function satisfies all the conditions of Rolle’s Theorem: it is continuous, it vanishes at the end points and has derivative at all points. Consequently, at some point such that ,
equals zero. But, . But, this contradicts hypothesis. Hence, the proof.