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 \( p(x) = x^3 - 2mx^2 + 16 \) is divisible by \( x + 2 \), we can use the Remainder Theorem. According to this theorem, if a polynomial \( p(x) \) is divisible by \( x + 2 \), then \( p(-2) = 0 \).

### Step-by-Step Solution:

1. **Set up the polynomial**:
\[ p(x) = x^3 - 2mx^2 + 16 \]

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

3. **Calculate \( (-2)^3 \)**:
\[ (-2)^3 = -8 \]

4. **Calculate \( (-2)^2 \)**:
\[ (-2)^2 = 4 \]

5. **Substitute these values into the polynomial**:
\[ p(-2) = -8 - 2m(4) + 16 \]

6. **Simplify the expression**:
\[ p(-2) = -8 - 8m + 16 \]
\[ p(-2) = 8 - 8m \]

7. **Set the polynomial equal to zero for divisibility**:
\[ 8 - 8m = 0 \]

8. **Solve for \( m \)**:
\[ 8 = 8m \]
\[ m = 1 \]

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