We are given the function:

$f(x) = x^3 + kx + 3$

Also, it's stated that $x - 3$ is a factor of $f(x)$, which means that $f(3) = 0$ (since if $x - 3$ is a factor of a polynomial, the polynomial must evaluate to zero at $x = 3$).

We can substitute $x = 3$ into the expression for $f(x)$ to find $k$.

Step 1: Substitute $x = 3$ into the equation

$f(3) = 3^3 + k(3) + 3$

This simplifies to:

$f(3) = 27 + 3k + 3 = 30 + 3k$

Since $x - 3$ is a factor, we know that $f(3) = 0$. Therefore, we set the equation equal to zero:

$30 + 3k = 0$

Step 2: Solve for $k$

$3k = -30$

$k = -10$

Thus, the value of $k$ is $\boxed{-10}$.

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If you have any questions or need further clarification, feel free to ask!

Here are some related questions you might find interesting:

1. How do you determine if a polynomial has a particular factor?
2. What does it mean for a polynomial to be divisible by a factor like $x - 3$?
3. How can you factorize a cubic polynomial like $x^3 + kx + 3$?
4. Can a quadratic polynomial also have a factor of the form $x - 3$?
5. What happens if you substitute a value into a polynomial and get zero?

Tip: Always remember to check the conditions given in the problem (like divisibility by $x - 3$) and use them to substitute into the polynomial to simplify solving for unknowns.