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The problem asks to find the value of \( k \) given that \( 3x^3 + kx + 10 \) is divisible by \( x + 2 \).

We are told that \( x + 2 \) is a factor of \( f(x) = 3x^3 + kx + 10 \). This means that when we substitute \( x = -2 \) into the polynomial \( f(x) \), the result must be zero (according to the factor theorem).

The function \( f(x) = 3x^3 + kx + 10 \).

Substituting \( x = -2 \):

\[
f(-2) = 3(-2)^3 + k(-2) + 10 = 0
\]

Simplify each term:

\[
3(-8) + (-2k) + 10 = 0
\]

\[
-24 - 2k + 10 = 0
\]

\[
-14 - 2k = 0
\]

Solve for \( k \):

\[
-2k = 14 \quad \Rightarrow \quad k = -7
\]

Thus, the value of \( k \) is \( \boxed{-7} \).

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Do you want more details on this solution? Here are some follow-up questions:

1. What is the Factor Theorem and how is it used to solve problems like this one?
2. How does synthetic division relate to finding factors of polynomials?
3. Can \( x + 2 \) have any other roots in this cubic equation?
4. How would the equation change if \( x + 2 \) was not a factor?
5. Can this type of method be used for polynomials of degree higher than three?

**Tip:** The Factor Theorem is very powerful for checking if a linear factor divides a polynomial without fully dividing the polynomial!

3x^3 + kx + 10, and x + 2 is a factor of f(x), then what is the value of k?

In this problem, find the value of k in the polynomial 3x^3 + kx + 10 when x + 2 is a factor. Using the Factor Theorem, learn how substituting x = -2 into the polynomial leads to the solution. This is suitable for students in Grades 10-12.

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