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The problem asks you to evaluate the polynomial function \( P(x) = 4x^4 - 4x^2 + 10x - 8 \) at \( x = 0.5 \) and round the result to the nearest hundredth.

Let's compute \( P(0.5) \).

1. First, substitute \( x = 0.5 \) into the function:
   \[
   P(0.5) = 4(0.5)^4 - 4(0.5)^2 + 10(0.5) - 8
   \]

2. Compute each term:
   - \( 4(0.5)^4 = 4 \times 0.0625 = 0.25 \)
   - \( -4(0.5)^2 = -4 \times 0.25 = -1 \)
   - \( 10(0.5) = 10 \times 0.5 = 5 \)
   - The constant \( -8 \) remains as \( -8 \).

3. Now, add these terms together:
   \[
   P(0.5) = 0.25 - 1 + 5 - 8
   \]
   \[
   P(0.5) = -3.75
   \]

Thus, \( P(0.5) = -3.75 \).

Would you like to explore how to apply the Intermediate Value Theorem next?

Here are 5 questions to expand the topic:
1. How do you apply the Intermediate Value Theorem in this problem?
2. What are the necessary conditions for using the Intermediate Value Theorem?
3. How can we approximate the zero of the function between 0.5 and 1?
4. What would \( P(1) \) be in this case?
5. How does this result help in determining whether there's a root between 0.5 and 1?

**Tip:** Always check the sign of the function at both bounds when applying the Intermediate Value Theorem to determine if a root exists in the interval.

Use the intermediate value theorem to show that the function P(x) = 4x^4 - 4x^2 + 10x - 8 has a zero between 0.5 and 1. Evaluate P(0.5).

Learn how to evaluate the polynomial P(x) = 4x^4 - 4x^2 + 10x - 8 at x = 0.5, using the Intermediate Value Theorem. This problem provides a step-by-step explanation and covers essential polynomial and calculus concepts. Suitable for high school students in Grades 10-12.

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