Question: Prove that the equation  has only one real root.

Proof:

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.

More later,

Nalin Pithwa