Use the Intermediate Value Theorem to show that the polynomial f\left(x\right)=x^{3}+x^{2}-2x+42 has a real zero between -5 and -2

To apply the Intermediate Value Theorem (IVT), we need to evaluate the polynomial \( f(x) = x^3 + x^2 - 2x + 42 \) at the endpoints of the interval \([-5, -2]\).

1. **Evaluate \( f(-5) \)**:
   \[
   f(-5) = (-5)^3 + (-5)^2 - 2(-5) + 42
   \]
   \[
   = -125 + 25 + 10 + 42 = -48
   \]

2. **Evaluate \( f(-2) \)**:
   \[
   f(-2) = (-2)^3 + (-2)^2 - 2(-2) + 42
   \]
   \[
   = -8 + 4 + 4 + 42 = 42
   \]

3. **Check the values**:
   - \( f(-5) = -48 \) (negative)
   - \( f(-2) = 42 \) (positive)

Since \( f(-5) < 0 \) and \( f(-2) > 0 \), and \( f(x) \) is a continuous polynomial function, the Intermediate Value Theorem guarantees that there is at least one real zero in the interval \([-5, -2]\).

### Summary
By the IVT, there is a real zero of the polynomial \( f(x) = x^3 + x^2 - 2x + 42 \) between -5 and -2.

Would you like more details on the Intermediate Value Theorem or related concepts? Here are some questions to consider:

1. What are the properties of continuous functions?
2. How does the IVT apply to other types of functions?
3. Can you provide another example of the IVT in action?
4. What are some numerical methods to find the actual zero?
5. How does the behavior of polynomial functions change with different degrees? 

**Tip**: Always check the function's continuity when applying the Intermediate Value Theorem, as it is a crucial condition for its validity.

Use the Intermediate Value Theorem to show that the polynomial f(x)=x^3+x^2-2x+42 has a real zero between -5 and -2.

Explore how to apply the Intermediate Value Theorem to the polynomial f(x) = x^3 + x^2 - 2x + 42, demonstrating the existence of a real zero between -5 and -2. This explanation includes detailed evaluations and the significance of continuity in polynomial functions. Ideal for high school students.

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