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To show that the equation \( x^4 + 7x^3 - 9 = 0 \) has at least one positive and one negative solution, we can apply the **Intermediate Value Theorem (IVT)**. The IVT states that if a continuous function \( f(x) \) takes on different signs at two points \( a \) and \( b \) (i.e., \( f(a) \) and \( f(b) \) have opposite signs), then there exists at least one root \( c \) in the interval \( (a, b) \) such that \( f(c) = 0 \).

Let's denote:
\[
f(x) = x^4 + 7x^3 - 9
\]
This function is continuous for all \( x \) because it is a polynomial.

### Steps to Use the Intermediate Value Theorem

1. **Evaluate \( f(x) \) at some positive and negative values to find sign changes:**

   - Let's choose \( x = 0 \):
     \[
     f(0) = 0^4 + 7 \cdot 0^3 - 9 = -9
     \]
     So, \( f(0) = -9 \), which is negative.

   - Now let's try a positive value, say \( x = 2 \):
     \[
     f(2) = 2^4 + 7 \cdot 2^3 - 9 = 16 + 56 - 9 = 63
     \]
     Here, \( f(2) = 63 \), which is positive.

2. **Apply the IVT for a positive solution:**

   Since \( f(0) = -9 \) and \( f(2) = 63 \), there is a sign change between \( x = 0 \) and \( x = 2 \). By the Intermediate Value Theorem, there must be at least one root \( c \) in the interval \( (0, 2) \), meaning there is at least one positive solution.

3. **Check a negative value, say \( x = -2 \):**
   \[
   f(-2) = (-2)^4 + 7 \cdot (-2)^3 - 9 = 16 - 56 - 9 = -49
   \]
   Here, \( f(-2) = -49 \), which is negative.

4. **Choose another negative value, say \( x = -3 \):**
   \[
   f(-3) = (-3)^4 + 7 \cdot (-3)^3 - 9 = 81 - 189 - 9 = -117
   \]
   Here, \( f(-3) = -117 \), which is also negative, indicating no sign change in this interval.

To verify

Show by the intermediate value theorem that the equation x^4 + 7x^3 - 9 = 0 has at least one positive and one negative solution.

Explore how to apply the Intermediate Value Theorem to prove that the polynomial equation x^4 + 7x^3 - 9 = 0 has at least one positive and one negative root. This problem illustrates the importance of continuous functions and polynomial behavior, ideal for Grades 10-12 students.

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