# 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}. 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,

$f^{'}(\xi)$ 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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