Fifth degree polynomial equation

Question: Prove that the equation 3x^{5}+15x-8 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 x_{1}<x_{2}. Then, in the interval [x_{1},x_{2}] 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 \xi such that x_{1}<\xi<x_{2},

f^{'}(\xi) equals zero. But, f^{'}(x)=15(x^{4}+1)>0. But, this contradicts hypothesis. Hence, the proof.

More later,

Nalin Pithwa

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