To determine the value of $k$ such that $(x+2)$ is a factor of the polynomial $2x^3 - 5x^2 + kx + 14$, we use the Factor Theorem. According to this theorem, $(x+2)$ is a factor if substituting $x = -2$ into the polynomial makes the result equal to zero.

Step 1: Substitute $x = -2$ into the polynomial:

The polynomial is: $P(x) = 2x^3 - 5x^2 + kx + 14$ Substitute $x = -2$: $P(-2) = 2(-2)^3 - 5(-2)^2 + k(-2) + 14$

Step 2: Simplify each term:

$2(-2)^3 = 2(-8) = -16$ $-5(-2)^2 = -5(4) = -20$ $k(-2) = -2k$ $+14 = 14$

Thus: $P(-2) = -16 - 20 - 2k + 14$

Step 3: Combine like terms:

$P(-2) = (-16 - 20 + 14) - 2k = -22 - 2k$

Step 4: Set $P(-2) = 0$ (since $x+2$ is a factor):

$-22 - 2k = 0$

Step 5: Solve for $k$:

$-2k = 22 \quad \Rightarrow \quad k = -11$

Final Answer:

The value of $k$ is: $\boxed{-11}$

Would you like further explanation or verification? Let me know!

Related Questions:

1. What is the Factor Theorem, and how does it work?
2. How do you determine if a polynomial is divisible by another polynomial?
3. Can this method be extended to higher-degree factors like $(x+a)^2$?
4. What is synthetic division, and how can it be used to verify the factorization?
5. How does the value of $k$ affect the overall shape of the polynomial graph?

Tip:

Always double-check factorization problems by substituting the solution back into the polynomial to verify correctness!