When the polynomial: x
3 + 2x
2
– kx – 16 is divided by (x + 2), there is a remainder of 
–2.
What is the value of k?
 
A. 6
B. 7
C. 8
D. 10


We are given the polynomial \( f(x) = x^3 + 2x^2 - kx - 16 \), and we are told that when it is divided by \( x + 2 \), the remainder is \(-2\). 

By the **Remainder Theorem**, if a polynomial \( f(x) \) is divided by \( x - a \), then the remainder is \( f(a) \). In this case, the divisor is \( x + 2 \), so we set \( a = -2 \). According to the Remainder Theorem:

\[
f(-2) = -2
\]

Now, we can substitute \( x = -2 \) into the polynomial and set it equal to \(-2\):

\[
f(-2) = (-2)^3 + 2(-2)^2 - k(-2) - 16 = -2
\]

Simplifying each term:

\[
(-2)^3 = -8, \quad 2(-2)^2 = 2(4) = 8, \quad -k(-2) = 2k, \quad -16 = -16
\]

Substitute these values into the equation:

\[
-8 + 8 + 2k - 16 = -2
\]

Simplifying the left-hand side:

\[
0 + 2k - 16 = -2
\]

Now solve for \( k \):

\[
2k - 16 = -2
\]

Add 16 to both sides:

\[
2k = 14
\]

Divide by 2:

\[
k = 7
\]

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

### Would you like more details on any step or have any questions?

Here are some related questions for further practice:
1. What does the Remainder Theorem state in general?
2. How do you apply synthetic division to find the remainder?
3. What happens if the remainder is zero? What does that tell us about the divisor?
4. If \( k = 7 \), what is the factorization of the polynomial?
5. How would you find the roots of the polynomial when \( k = 7 \)?

**Tip:** When dividing polynomials by binomials, the Remainder Theorem can quickly help you find the remainder without full division!

When the polynomial: x^3 + 2x^2 – kx – 16 is divided by (x + 2), there is a remainder of -2. What is the value of k? A. 6 B. 7 C. 8 D. 10

Solve the problem of finding the value of k in the polynomial x^3 + 2x^2 - kx - 16, when divided by (x + 2) with a remainder of -2. This solution uses the Remainder Theorem and involves key algebraic concepts suitable for Grades 9-11.

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