The value of k so that ` x^4 -3x^3 +5x^2 -33 x +k` is divisible by ` x^2 -5x +6 ` is

Options

A. 45
B. 48
C. 51
D. 54

Answer

To find the value of \( k \) such that the polynomial \( x^4 - 3x^3 + 5x^2 - 33x + k \) is divisible by \( x^2 - 5x + 6 \), we will follow these steps:

### Step 1: Factor the divisor
First, we factor the quadratic polynomial \( x^2 - 5x + 6 \):
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]
This tells us that the roots of the polynomial are \( x = 2 \) and \( x = 3 \).

### Step 2: Evaluate the polynomial at the roots
For the polynomial \( P(x) = x^4 - 3x^3 + 5x^2 - 33x + k \) to be divisible by \( x^2 - 5x + 6 \), it must equal zero at both roots.

#### Evaluating at \( x = 2 \):
\[ P(2) = 2^4 - 3(2^3) + 5(2^2) - 33(2) + k \]
Calculating each term:
\[ = 16 - 24 + 20 - 66 + k \]
\[ = -54 + k \]
Setting \( P(2) = 0 \):
\[ -54 + k = 0 \implies k = 54 \]

#### Evaluating at \( x = 3 \):
Now, we check at \( x = 3 \) to ensure consistency:
\[ P(3) = 3^4 - 3(3^3) + 5(3^2) - 33(3) + k \]
Calculating each term:
\[ = 81 - 81 + 45 - 99 + k \]
\[ = -54 + k \]
Setting \( P(3) = 0 \):
\[ -54 + k = 0 \implies k = 54 \]

### Conclusion
Both evaluations give us the same result, confirming that the value of \( k \) is \( 54 \).

Thus, the value of \( k \) such that \( x^4 - 3x^3 + 5x^2 - 33x + k \) is divisible by \( x^2 - 5x + 6 \) is:
\[ \boxed{54} \]