
Concept explainers
Maximum Number of real positive or negative real zeros of the given polynomial function using Descartes’ Rule of signs and the potential rational zeros of each polynomial function using rational zeros theorem.

Answer to Problem 25AYU
The degree of the polynomial is 3. Therefore the number of real zeros by real zero theorem can be at most .
Therefore there will be 1 positive real zero.
Therefore there can be two negative real zeros or no negative real zero.
The rational zeros .
Explanation of Solution
Given:
The degree of the polynomial is 3. Therefore the number of real zeros by real zero theorem can be at most .
to
There is 1 variation of sign of nonzero coefficient of by Descartes’ Rule of signs;
Number of positive real zeros number of variations of signs of or number of variations less than even integer.
Therefore there will be 1 positive real zero or no positive real zero.
Let us consider by replacing by .
to to
There are two variations of signs of nonzero coefficient of by Descartes’ Rule of signs;
Number of negative real zeros number of variations of signs or number of variations less than even integer.
Therefore there can be two negative real zeros or no negative real zero.
Rational zeros theorem provides information about the rational zeros of a polynomial function with integer coefficients.
If in its lowest terms is a rational zero of , then is a factor of and is the factor of .
Here and .
Zeros of , .
Zeros of 2, .
The potential rational zeros .
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