Zeros of a Polynomial or Rational Function and Its Derivative

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This Demonstration shows the connection between the real zeros of a function and those of its derivative . It is also shown that for rational functions, the asymptotes of the function match those of the derivative.

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When is a polynomial, a zero of of multiplicity is a zero of with multiplicity .

If

where and the are distinct, then is the number of distinct zeros, and

,

where introduces additional zeros.

In the case of a rational function, the rule becomes:

,

where , , the are distinct, the are distinct, and

,

where is the number of distinct zeros of the numerator and is the number of distinct zeros of the denominator.

When the rule becomes:

,

where

, and

where is the number of distinct zeros of the numerator and is the number of distinct zeros of the denominator.

That is to say, the derivative will have one fewer zero and the asymptotes will remain.

The zeros of the are highlighted with , while the zeros of the are highlighted with ; in the case of a rational function, the vertical asymptotes are marked by .

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Contributed by: D. Meliga and S. Z. Lavagnino (November 2017)
Additional contributions by: E. Perano
Open content licensed under CC BY-NC-SA


Snapshots


Details

Snapshot 2: shows a rational function and its derivative. Besides the zeros of and , we can find the vertical asymptotes of , which can also be found in .

Reference

[1] E. Perano, Il teorema di Rolle, Torino, Italy: Edizioni Cortina, 2017.



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