To solve this problem using the Factor Theorem, follow these steps:

1. The Factor Theorem states that if $P(x)$ is a polynomial and $x - c$ is a factor of $P(x)$, then $P(c) = 0$. In other words, if the remainder when dividing $P(x)$ by $x - c$ is zero, then $x - c$ is a factor.

2. Here, the binomial is $3x - 1$. First, rewrite this binomial in the form $x - c$: $3x - 1 = 0 \implies x = \frac{1}{3}.$ So, we need to check if $P\left(\frac{1}{3}\right) = 0$.

3. The polynomial is $P(x) = 3x^4 - 7x^3 + x - 2$.

4. Now, substitute $x = \frac{1}{3}$ into $P(x)$: $P\left(\frac{1}{3}\right) = 3\left(\frac{1}{3}\right)^4 - 7\left(\frac{1}{3}\right)^3 + \left(\frac{1}{3}\right) - 2.$

Let's calculate the value of $P\left(\frac{1}{3}\right)$.After calculating $P\left(\frac{1}{3}\right)$, we get approximately -1.89.

Since $P\left(\frac{1}{3}\right) \neq 0$, this means that $3x - 1$ is not a factor of the polynomial $P(x) = 3x^4 - 7x^3 + x - 2$.

The remainder when dividing $P(x)$ by $3x - 1$ is approximately -1.89.

Would you like to explore more details on this, or ask any further questions?

Here are some related questions for you:

1. How is the Remainder Theorem connected to the Factor Theorem?
2. What are the steps to factor a polynomial if no obvious factors are found?
3. How do synthetic division and the Factor Theorem work together in polynomial division?
4. How can we find all possible rational roots of a polynomial?
5. What is the difference between factoring by grouping and using the Factor Theorem?

Tip: If a binomial isn't a factor, the remainder tells you how far off the exact division is.