When `x^3+2x^2-kx+4` . is divided by x - 2. the remainder is k. Find the value of constant k.

Answer

To solve the problem, we need to find the value of the constant \( k \) such that when the polynomial \( f(x) = x^3 + 2x^2 - kx + 4 \) is divided by \( x - 2 \), the remainder is \( k \).

### Step-by-Step Solution:

1. **Identify the Polynomial**:
Let \( f(x) = x^3 + 2x^2 - kx + 4 \).

2. **Use the Remainder Theorem**:
According to the Remainder Theorem, when a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). Here, we are dividing by \( x - 2 \), so we will evaluate \( f(2) \).

3. **Calculate \( f(2) \)**:
Substitute \( x = 2 \) into the polynomial:
\[ f(2) = 2^3 + 2(2^2) - k(2) + 4 \]
Simplifying this:
\[ f(2) = 8 + 2(4) - 2k + 4 \]
\[ f(2) = 8 + 8 - 2k + 4 \]
\[ f(2) = 20 - 2k \]

4. **Set the Remainder Equal to \( k \)**:
According to the problem, the remainder when dividing \( f(x) \) by \( x - 2 \) is \( k \). Therefore, we set up the equation:
\[ 20 - 2k = k \]

5. **Solve for \( k \)**:
Rearranging the equation:
\[ 20 = k + 2k \]
\[ 20 = 3k \]
Dividing both sides by 3:
\[ k = \frac{20}{3} \]

### Final Answer:
The value of the constant \( k \) is \( \frac{20}{3} \).

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