If the polynomial `x^(3)+2x^(2)-alphax-12` is divided by `(x-4)` the remainder is `52` . Find the value of `alpha` .

Options

A. `11//2`
B. `-5`
C. `8`
D. `-8`

Answer

To find the value of α in the polynomial \( P(x) = x^3 + 2x^2 - \alpha x - 12 \) given that the remainder when divided by \( x - 4 \) is 52, we can follow these steps:

### Step 1: Apply the Remainder Theorem
According to the Remainder Theorem, when a polynomial \( P(x) \) is divided by \( x - c \), the remainder is \( P(c) \). In this case, since we are dividing by \( x - 4 \), we will evaluate \( P(4) \).

### Step 2: Substitute \( x = 4 \) into the polynomial
We substitute \( x = 4 \) into the polynomial:
\[ P(4) = 4^3 + 2(4^2) - \alpha(4) - 12 \]

### Step 3: Calculate \( P(4) \)
Calculating each term:
- \( 4^3 = 64 \)
- \( 4^2 = 16 \), so \( 2(4^2) = 2 \times 16 = 32 \)

Now substituting these values into the polynomial:
\[ P(4) = 64 + 32 - 4\alpha - 12 \]

### Step 4: Simplify the expression
Now, we simplify:
\[ P(4) = 64 + 32 - 12 - 4\alpha = 84 - 4\alpha \]

### Step 5: Set the equation equal to the remainder
We know from the problem statement that \( P(4) = 52 \). Therefore, we set up the equation:
\[ 84 - 4\alpha = 52 \]

### Step 6: Solve for \( \alpha \)
Now, we solve for \( \alpha \):
1. Subtract 84 from both sides:
\[ -4\alpha = 52 - 84 \]
\[ -4\alpha = -32 \]

2. Divide both sides by -4:
\[ \alpha = \frac{-32}{-4} = 8 \]

### Conclusion
Thus, the value of \( \alpha \) is \( 8 \).