To determine the value of $k$ so that $x - 3$ is a factor of $x^3 - 3x^2 + x + k$, we can use the Factor Theorem. This theorem states that if $x - c$ is a factor of a polynomial $f(x)$, then $f(c) = 0$.

Here, $c = 3$. We need to find $k$ such that:

$f(3) = 3^3 - 3(3^2) + 3 + k = 0$

Calculating $f(3)$:

$f(3) = 27 - 27 + 3 + k$ $f(3) = 3 + k$

Setting this equal to zero:

$3 + k = 0$

Solving for $k$:

$k = -3$

Thus, the value of $k$ is $-3$.

Would you like more details on this method, or do you have any questions? Here are some related questions:

1. What is the Factor Theorem?
2. How can we verify that $x - 3$ is indeed a factor?
3. What are other methods to find factors of polynomials?
4. Can you explain polynomial long division?
5. What happens if we choose a different value for $k$?

Tip: Always check your work by substituting back to see if the factor holds!