Determine the possible number of positive and negative real zeros of the following: 1. P(x)=x⁵-3x⁴+3x³-9x²+2x

To determine the possible number of positive and negative real zeros of the polynomial function \( P(x) = x^5 - 3x^4 + 3x^3 - 9x^2 + 2x \), we can use **Descartes' Rule of Signs**. This rule helps predict the number of positive and negative real zeros by examining the sign changes in \( P(x) \) and \( P(-x) \).

### Step 1: Positive Real Zeros
1. **Analyze \( P(x) \):**
   \[
   P(x) = x^5 - 3x^4 + 3x^3 - 9x^2 + 2x
   \]
2. **Identify the sign changes:**
   - From \( +x^5 \) to \( -3x^4 \): **1 sign change**
   - From \( -3x^4 \) to \( +3x^3 \): **1 sign change**
   - From \( +3x^3 \) to \( -9x^2 \): **1 sign change**
   - From \( -9x^2 \) to \( +2x \): **1 sign change**

   So, there are **4 sign changes** in \( P(x) \).

3. **Conclusion for positive real zeros:**
   According to Descartes' Rule of Signs, the number of positive real zeros of \( P(x) \) can be \( 4 \), \( 2 \), or \( 0 \), decreasing by 2 each time.

### Step 2: Negative Real Zeros
1. **Analyze \( P(-x) \):**
   - Substitute \( -x \) for \( x \) in \( P(x) \):
     \[
     P(-x) = (-x)^5 - 3(-x)^4 + 3(-x)^3 - 9(-x)^2 + 2(-x)
     \]
     Simplify each term:
     \[
     P(-x) = -x^5 - 3x^4 - 3x^3 - 9x^2 - 2x
     \]
     This simplifies to:
     \[
     P(-x) = -x^5 - 3x^4 - 3x^3 - 9x^2 - 2x
     \]
2. **Identify the sign changes:**
   Since all terms in \( P(-x) \) are negative, there are **no sign changes**.

3. **Conclusion for negative real zeros:**
   There are **no negative real zeros** of \( P(x) \).

### Summary
- Possible number of **positive real zeros**: \( 4 \), \( 2 \), or \( 0 \).
- Possible number of **negative real zeros**: \( 0 \).

Would you like a more detailed explanation of Descartes' Rule of Signs or any follow-up questions?

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#### Follow-up Questions
1. How does Descartes' Rule of Signs apply to a polynomial with complex roots?
2. How do we determine the total number of roots for a polynomial?
3. What can we infer about the number of complex roots in this polynomial?
4. How would we determine the exact number of positive real roots?
5. Can this polynomial have any repeated real roots?

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#### Tip
For polynomials with higher degrees, a combination of Descartes' Rule of Signs and other methods, such as synthetic division or graphing, can be very helpful to find all real roots.

Determine the possible number of positive and negative real zeros of the following: P(x)=x⁵-3x⁴+3x³-9x²+2x

Explore how to determine the positive and negative real zeros of the polynomial P(x) = x⁵ - 3x⁴ + 3x³ - 9x² + 2x using Descartes' Rule of Signs. This analysis reveals potential zeros and is suitable for advanced high school mathematics students.

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