Precalculus : Determine the Number of Positive and Negative Real Zeros of a Polynomial Using Descartes' Rule of Signs

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #2 : Descartes' Rule, Intermediate Value Theorem, Sum And Product Of Zeros

Determine the possible number of Positive and Negative Real Zeros of the polynomial using Descartes' Rule of Signs

Possible Answers:

Positive zeroes: 

Negative zeroes: 

Positive zeroes: 

Negative zeroes: 

Positive zeroes: 

Negative zeroes: 

Positive zeroes: 

Negative zeroes: 

Correct answer:

Positive zeroes: 

Negative zeroes: 

Explanation:

In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial. The number of real zeroes can then be any positive difference of that number and a positive multiple of two.

For the function

there are four sign changes. As such, the number of positive real zeroes can be

In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial after substituting for  The number of real zeroes can then be any positive difference of that number and a positive multiple of two.

After substituting  we get

and there is one sign change.

As such, there can only be one negative root.

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