To solve the given problem, let's proceed step by step:

Problem:

We are given $f(x) = 3x^3 + 7x + k$, and it is stated that $x + 1$ is a factor of $f(x)$. We need to find the value of $k$.

Concept:

If $x + 1$ is a factor of $f(x)$, then by the factor theorem, $f(-1) = 0$.

Steps:

1. Substitute $x = -1$ into $f(x) = 3x^3 + 7x + k$. $f(-1) = 3(-1)^3 + 7(-1) + k$
2. Simplify: $f(-1) = 3(-1) + 7(-1) + k = -3 - 7 + k$ $f(-1) = -10 + k$
3. Since $x + 1$ is a factor, $f(-1) = 0$. Thus: $-10 + k = 0$
4. Solve for $k$: $k = 10$

Final Answer:

The value of $k$ is $\boxed{10}$.

Do you want more details or have any further questions?

Related Questions:

1. What is the factor theorem, and how is it applied to polynomial functions?
2. If $x - 2$ were a factor instead of $x + 1$, how would we calculate $k$?
3. How can we verify that $f(x) = 3x^3 + 7x + 10$ has $x + 1$ as a factor?
4. What is the remainder theorem, and how is it related to the factor theorem?
5. What would happen if there were no value of $k$ that satisfied the condition?

Tip:

Always double-check factorization by substituting back into the polynomial to confirm your solution satisfies the given condition.