To determine which of the given expressions is a factor of the polynomial $P(x) = 2x^3 - 7x + 3$, we can use the Factor Theorem.

The Factor Theorem states that for a polynomial $P(x)$, if $P(c) = 0$, then $x - c$ is a factor of $P(x)$.

Let's check each option:

1. For $x - 3$: Set $x = 3$ in $P(x)$: $P(3) = 2(3)^3 - 7(3) + 3 = 2(27) - 21 + 3 = 54 - 21 + 3 = 36$ Since $P(3) \neq 0$, $x - 3$ is not a factor.

2. For $x + 1$: Set $x = -1$ in $P(x)$: $P(-1) = 2(-1)^3 - 7(-1) + 3 = 2(-1) + 7 + 3 = -2 + 7 + 3 = 8$ Since $P(-1) \neq 0$, $x + 1$ is not a factor.

3. For $3x - 2$: First, express this as $x = \frac{2}{3}$. Now, substitute $x = \frac{2}{3}$ into $P(x)$: $P\left( \frac{2}{3} \right) = 2\left( \frac{2}{3} \right)^3 - 7\left( \frac{2}{3} \right) + 3 = 2\left( \frac{8}{27} \right) - \frac{14}{3} + 3$ Simplifying: $P\left( \frac{2}{3} \right) = \frac{16}{27} - \frac{126}{27} + \frac{81}{27} = \frac{16 - 126 + 81}{27} = \frac{-29}{27}$ Since $P\left( \frac{2}{3} \right) \neq 0$, $3x - 2$ is not a factor.

Conclusion:

None of the given expressions is a factor of the polynomial $P(x) = 2x^3 - 7x + 3$.

Would you like a detailed explanation of these steps or have any questions?

Related Questions:

1. What is the Factor Theorem, and how does it work?
2. How can we verify if a higher-degree polynomial has multiple factors?
3. Can synthetic division be used to factor polynomials efficiently?
4. What is the relationship between the Remainder Theorem and the Factor Theorem?
5. How can we factor polynomials when none of the simple factor tests work?

Tip: Always remember, if $P(c) = 0$, then $x - c$ is a factor of $P(x)$.