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.