Using Descartes law of Signs, what is the maximum possible number of Negative Zeros for the polynomial f(x)= 5x^5 +10x^4 -37x^3 -74x^2 +14x +28?

To determine the maximum possible number of negative zeros of the polynomial \( f(x) = 5x^5 + 10x^4 - 37x^3 - 74x^2 + 14x + 28 \), we can apply Descartes' Rule of Signs.

1. **Identify \( f(-x) \):** First, we substitute \( -x \) for \( x \) in \( f(x) \) to determine how many sign changes occur in \( f(-x) \).

   \[
   f(-x) = 5(-x)^5 + 10(-x)^4 - 37(-x)^3 - 74(-x)^2 + 14(-x) + 28
   \]

   Simplify each term:
   - \( 5(-x)^5 = -5x^5 \)
   - \( 10(-x)^4 = 10x^4 \)
   - \( -37(-x)^3 = 37x^3 \)
   - \( -74(-x)^2 = -74x^2 \)
   - \( 14(-x) = -14x \)
   - \( 28 \) remains \( 28 \)

   Therefore,

   \[
   f(-x) = -5x^5 + 10x^4 + 37x^3 - 74x^2 - 14x + 28
   \]

2. **Count the sign changes in \( f(-x) \):**
   Now we look at the sequence of terms in \( f(-x) \):
   - \( -5x^5, 10x^4, 37x^3, -74x^2, -14x, 28 \)

   Observing the signs:
   - From \( -5x^5 \) to \( 10x^4 \): 1st sign change
   - From \( 10x^4 \) to \( 37x^3 \): no change
   - From \( 37x^3 \) to \( -74x^2 \): 2nd sign change
   - From \( -74x^2 \) to \( -14x \): no change
   - From \( -14x \) to \( 28 \): 3rd sign change

   Thus, there are **3 sign changes** in \( f(-x) \).

3. **Interpret using Descartes' Rule of Signs:**
   Descartes' Rule of Signs states that the number of negative real zeros of \( f(x) \) is equal to the number of sign changes in \( f(-x) \) or less than that by an even number. Therefore, the possible number of negative real zeros is:

   \[
   3 \text{ or } 3 - 2 = 1
   \]

### Conclusion
The maximum possible number of negative zeros for \( f(x) \) is **3**.

Would you like a detailed explanation of Descartes' Rule of Signs or further assistance?

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### Follow-up Questions:
1. How would you find the maximum possible number of positive zeros for this polynomial?
2. What are the possible numbers of positive, negative, or complex zeros for a fifth-degree polynomial?
3. How could you determine if any negative zeros actually exist?
4. How would you apply Descartes' Rule of Signs to a polynomial with only even or only odd terms?
5. How would synthetic division help in finding actual zeros of this polynomial?

### Tip:
Descartes' Rule of Signs gives possible numbers of positive or negative zeros but doesn't confirm actual zeros. For actual roots, use methods like synthetic division, factoring, or graphing.

Using Descartes' Law of Signs, what is the maximum possible number of Negative Zeros for the polynomial f(x)= 5x^5 +10x^4 -37x^3 -74x^2 +14x +28?

Learn how to determine the maximum possible number of negative zeros for the polynomial f(x) = 5x^5 +10x^4 -37x^3 -74x^2 +14x +28 using Descartes' Rule of Signs. This solution provides a detailed analysis, identifying sign changes in the polynomial and applying the rule to find possible negative real zeros. Suitable for high school students in Grades 10-12.

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