For what value of m is `x^(3)-2mx^(2)+16` Divisible by x+2 ?

Options

A. -1
B. -2
C. 1
D. 8

Explanation

To determine the value of \( m \) for which the polynomial \( x^3 - 2mx^2 + 16 \) is divisible by \( x + 2 \), we can use the Remainder Theorem. According to this theorem, if a polynomial \( f(x) \) is divisible by \( x - a \), then \( f(a) = 0 \).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
We have the polynomial \( f(x) = x^3 - 2mx^2 + 16 \) and the divisor \( x + 2 \).

2. **Set up the equation using the Remainder Theorem**:
Since \( x + 2 \) is a factor, we set \( x + 2 = 0 \) which gives us \( x = -2 \). We need to find \( m \) such that \( f(-2) = 0 \).

3. **Substitute \( x = -2 \) into the polynomial**:
\[ f(-2) = (-2)^3 - 2m(-2)^2 + 16 \]

4. **Calculate \( f(-2) \)**:
\[ f(-2) = -8 - 2m(4) + 16 \]
Simplifying this gives:
\[ f(-2) = -8 - 8m + 16 \]
\[ f(-2) = 8 - 8m \]

5. **Set the equation to zero**:
Since \( f(-2) = 0 \):
\[ 8 - 8m = 0 \]

6. **Solve for \( m \)**:
Rearranging the equation gives:
\[ 8m = 8 \]
Dividing both sides by 8:
\[ m = 1 \]

### Conclusion:
The value of \( m \) for which the polynomial \( x^3 - 2mx^2 + 16 \) is divisible by \( x + 2 \) is \( m = 1 \).