Analysis Basics

Take 10 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel to ask then on the forum.

Question

We denote by a polynomial.

Prove that equation has only a finite number of roots.

Indice

Apply successively several times Rolle's theorem, until the polynomial disappears.

Solution

Let be the degree of the polynomial. We set the number of roots of equation is equal to the number of zeros of application

Let us assume that application has zeros, from the result of the previous exercise, this implies that application takes the zero value at least once.

However the derivative of order of a polynomial of degree is zero, hence which is never zero.

The equation has therefore at most solutions.

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