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For each of the following functions, (a) Use Descartes' Rule of Sign to determine all the possible positive-negative-complex configurations of the zeros. (b) Assuming all the zeros are real, determine

Answer 1

Using Descartes' Rule of Signs, you can determine the possible configurations of zeros for a function.

Step-by-step explanation:

In order to determine the zero configurations of the given functions, we can use Descartes' Rule of Signs. Let's take an example function: f(x) = x^3 + 2x^2 - 4x - 4.

1. Count the number of sign changes in f(x): There are two sign changes.
2. Count the number of sign changes in f(-x): There are no sign changes.
3. The number of positive-real zeros is either equal to the number of sign changes or is less by an even number: In our example, there are two or no positive-real zeros.
4. The number of negative-real zeros is found by substituting -x in the function and counting the sign changes. There are no sign changes in our example function, so no negative-real zeros.
5. The number of complex zeros is found by subtracting the number of positive-real zeros from the number of sign changes. In our example, we have two complex zeros.

Using this method, you can determine the possible configurations of zeros for other functions as well.