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To determine if \( x - 1 \) is a factor of \( P(x) = -2x^3 + 3x^2 - 1 \) using the **Factor Theorem**, we need to evaluate \( P(1) \).

The **Factor Theorem** states that if \( x - c \) is a factor of \( P(x) \), then \( P(c) = 0 \). In this case, \( c = 1 \), so let's substitute \( x = 1 \) into \( P(x) \) and evaluate it:

\[
P(1) = -2(1)^3 + 3(1)^2 - 1
\]
\[
= -2 \cdot 1 + 3 \cdot 1 - 1
\]
\[
= -2 + 3 - 1
\]
\[
= 0
\]

Since \( P(1) = 0 \), according to the Factor Theorem, \( x - 1 \) **is a factor of** \( P(x) \).

### Answer:
- **Select**: \( x - 1 \) is a factor of \( P(x) \).

Would you like further details on how the Factor Theorem works or have any questions?

Here are five related questions to explore further:
1. How would you determine if \( x + 2 \) is a factor of another polynomial?
2. What are the steps to fully factorize \( P(x) = -2x^3 + 3x^2 - 1 \)?
3. Can the Factor Theorem be used to find multiple factors of a polynomial?
4. What if \( P(1) \) resulted in a non-zero value; what would that imply?
5. How does synthetic division help in factorizing polynomials?

**Tip**: Always substitute carefully when using the Factor Theorem to avoid errors in calculations.

Use the Factor Theorem to determine whether \( x - 1 \) is a factor of \( P(x) = -2x^3 + 3x^2 - 1 \). Specifically, evaluate \( P \) at the proper value, and then determine whether \( x - 1 \) is a factor.

Learn how to use the Factor Theorem to determine if \( x - 1 \) is a factor of \( P(x) = -2x^3 + 3x^2 - 1 \). This solution includes evaluating \( P(1) \) to see if it equals zero, indicating that \( x - 1 \) is a factor. Perfect for students in Grades 10-12 studying polynomial functions.

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