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To solve the problem using **Descartes' Rule of Signs**, we need to analyze the given polynomials in terms of sign changes for \( P(x) \) (positive real roots) and \( P(-x) \) (negative real roots). Here's how the rule works:

1. **For positive real roots:** Count the number of sign changes in the polynomial \( P(x) \).
2. **For negative real roots:** Substitute \( x \) with \( -x \) in \( P(x) \), and then count the number of sign changes in the new expression.

Let's go through each polynomial one by one:

### 1. \( P(x) = x^4 + 4x^3 - 7x^2 - 22x + 24 \)

- **Positive roots:**
  - \( P(x) = x^4 + 4x^3 - 7x^2 - 22x + 24 \)
  - Signs: \( +, +, -, -, + \) → 2 sign changes.
  - Possible positive real roots: 2 or 0.

- **Negative roots:**
  - \( P(-x) = x^4 - 4x^3 - 7x^2 + 22x + 24 \)
  - Signs: \( +, -, -, +, + \) → 2 sign changes.
  - Possible negative real roots: 2 or 0.

---

### 2. \( P(x) = x^4 + 6x^3 + 5x^2 - 26x - 24 \)

- **Positive roots:**
  - \( P(x) = x^4 + 6x^3 + 5x^2 - 26x - 24 \)
  - Signs: \( +, +, +, -, - \) → 2 sign changes.
  - Possible positive real roots: 2 or 0.

- **Negative roots:**
  - \( P(-x) = x^4 - 6x^3 + 5x^2 + 26x - 24 \)
  - Signs: \( +, -, +, +, - \) → 3 sign changes.
  - Possible negative real roots: 3 or 1.

---

### 3. \( P(x) = 2x^5 + 7x^4 - 18x^3 - 37x^2 + 70x - 24 \)

- **Positive roots:**
  - \( P(x) = 2x^5 + 7x^4 - 18x^3 - 37x^2 + 70x - 24 \)
  - Signs: \( +, +, -, -, +, - \) → 3 sign changes.
  - Possible positive real roots: 3 or 1.

- **Negative roots:**
  - \( P(-x) = -2x^5 + 7x^4 + 18x^3 - 37x^2 - 70x - 24 \)
  - Signs: \( -, +, +, -, -, - \) → 3 sign changes.
  - Possible negative real roots: 3 or 1.

---

### 4. \( P(x) = 2x^4 + 9x^3 - 9x^2 - 46x + 24 \)

- **Positive roots:**
  - \( P(x) = 2x^4 + 9x^3 - 9x^2 - 46x + 24 \)
  - Signs: \( +, +, -, -, + \) → 2 sign changes.
  - Possible positive real roots: 2 or 0.

- **Negative roots:**
  - \( P(-x) = 2x^4 - 9x^3 - 9x^2 + 46x + 24 \)
  - Signs: \( +, -, -, +, + \) → 2 sign changes.
  - Possible negative real roots: 2 or 0.

---

### 5. \( P(x) = 2x^6 + 9x^5 - 7x^4 - 37x^3 + 15x^2 - 46x + 24 \)

- **Positive roots:**
  - \( P(x) = 2x^6 + 9x^5 - 7x^4 - 37x^3 + 15x^2 - 46x + 24 \)
  - Signs: \( +, +, -, -, +, -, + \) → 4 sign changes.
  - Possible positive real roots: 4, 2, or 0.

- **Negative roots:**
  - \( P(-x) = 2x^6 - 9x^5 - 7x^4 + 37x^3 + 15x^2 + 46x + 24 \)
  - Signs: \( +, -, -, +, +, +, + \) → 2 sign changes.
  - Possible negative real roots: 2 or 0.

---

### 6. \( P(x) = x^6 - 5x^5 + 3x^4 - 5x^3 - 6x^2 - 4x - 8 \)

- **Positive roots:**
  - \( P(x) = x^6 - 5x^5 + 3x^4 - 5x^3 - 6x^2 - 4x - 8 \)
  - Signs: \( +, -, +, -, -, -, - \) → 5 sign changes.
  - Possible positive real roots: 5, 3, or 1.

- **Negative roots:**
  - \( P(-x) = x^6 + 5x^5 + 3x^4 + 5x^3 - 6x^2 + 4x - 8 \)
  - Signs: \( +, +, +, +, -, +, - \) → 2 sign changes.
  - Possible negative real roots: 2 or 0.

---

Do you want more details on how each step was performed or have any specific questions?

Here are some related questions for practice:
1. How does Descartes' Rule of Signs apply to non-polynomial functions?
2. Can a polynomial have more positive roots than the number of sign changes in its terms?
3. Why does Descartes' Rule of Signs only give the *possible* number of real roots, not the exact number?
4. How does the Fundamental Theorem of Algebra relate to finding the number of roots in polynomials?
5. What happens when a polynomial has no sign changes? Does it mean there are no real roots?

**Tip:** When using Descartes' Rule of Signs, remember to also use synthetic division or other methods to fully determine the actual roots!

Use Descartes' Rule of Signs to state the possible number of positive and negative real zeros of each polynomial function.

1. P(x) = x^4 + 4x^3 - 7x^2 - 22x + 24
2. P(x) = x^4 + 6x^3 + 5x^2 - 26x - 24
3. P(x) = 2x^5 + 7x^4 - 18x^3 - 37x^2 + 70x - 24
4. P(x) = 2x^4 + 9x^3 - 9x^2 - 46x + 24
5. P(x) = 2x^6 + 9x^5 - 7x^4 - 37x^3 + 15x^2 - 46x + 24
6. P(x) = x^6 - 5x^5 + 3x^4 - 5x^3 - 6x^2 - 4x - 8

Explore how to use Descartes' Rule of Signs to determine the possible number of positive and negative real zeros for various polynomial functions. This problem covers key algebraic concepts and is suitable for students in Grades 10-12.

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