Use the intermediate value theorem to show that the polynomial function has a real zero between the numbers given.

x Superscript 4 Baseline minus 3 x cubed minus 15 x squared plus 24 x minus 100x4−3x3−15x2+24x − 100​;

negative 4−4

and negative 3

To apply the Intermediate Value Theorem (IVT), we first need to verify that the polynomial function has different signs at \( x = -4 \) and \( x = -3 \). The IVT states that if a continuous function \( f(x) \) changes sign over an interval \([a, b]\), then there exists at least one real zero in that interval.

Given:
\[
f(x) = x^4 - 3x^3 - 15x^2 + 24x - 100
\]

We need to calculate \( f(-4) \) and \( f(-3) \) and check their signs.

### Step 1: Calculate \( f(-4) \)
\[
f(-4) = (-4)^4 - 3(-4)^3 - 15(-4)^2 + 24(-4) - 100
\]
\[
= 256 + 192 - 240 - 96 - 100
\]
\[
= 256 + 192 - 240 - 96 - 100 = 12
\]

So, \( f(-4) = 12 \).

### Step 2: Calculate \( f(-3) \)
\[
f(-3) = (-3)^4 - 3(-3)^3 - 15(-3)^2 + 24(-3) - 100
\]
\[
= 81 + 81 - 135 - 72 - 100
\]
\[
= 81 + 81 - 135 - 72 - 100 = -145
\]

So, \( f(-3) = -145 \).

### Step 3: Apply the Intermediate Value Theorem
We observe that:
\[
f(-4) = 12 > 0 \quad \text{and} \quad f(-3) = -145 < 0
\]

Since \( f(x) \) is continuous (it’s a polynomial), and the function changes sign between \( x = -4 \) and \( x = -3 \), the IVT guarantees that there is at least one real zero in the interval \((-4, -3)\).

### Conclusion
By the Intermediate Value Theorem, the polynomial function \( f(x) = x^4 - 3x^3 - 15x^2 + 24x - 100 \) has a real zero in the interval \((-4, -3)\).

Would you like more details on this method or have any questions?

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**Relative Questions:**
1. How do we know the Intermediate Value Theorem applies to all polynomials?
2. Can there be more than one real zero in the interval \((-4, -3)\)?
3. How does the Intermediate Value Theorem work if the function does not change signs?
4. What happens if both \( f(-4) \) and \( f(-3) \) are positive?
5. How can we approximate the zero between \(-4\) and \(-3\) more precisely?

**Tip:** The Intermediate Value Theorem is often most helpful in confirming the existence of roots, not their exact location—other techniques can pinpoint them more precisely.

Use the intermediate value theorem to show that the polynomial function has a real zero between the numbers given: x^4 - 3x^3 - 15x^2 + 24x - 100; -4 and -3.

Explore how to apply the Intermediate Value Theorem to determine if the polynomial function f(x) = x^4 - 3x^3 - 15x^2 + 24x - 100 has a real zero between x = -4 and x = -3. This solution involves step-by-step calculations and verifies the presence of a real zero in the interval, suitable for students in Grades 11-12.

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