When `(x^3-2x^2+px-q)` is divided by `(x^2-2x-3)` the remainder is (x-6)The values of p and q are:

Options

A. p=-2,q=-6
B. p=2,q=-6
C. p=-2,q=6
D. p=2,q=6

Answer

To solve the problem, we need to find the values of \( p \) and \( q \) such that when the polynomial \( (x^3 - 2x^2 + px - q) \) is divided by \( (x^2 - 2x - 3) \), the remainder is \( (x - 6) \).

### Step 1: Set up the polynomial division
We start by expressing the polynomial division:
\[ f(x) = x^3 - 2x^2 + px - q \]
When divided by \( g(x) = x^2 - 2x - 3 \), we know that:
\[ f(x) = g(x) \cdot Q(x) + R(x) \]
where \( R(x) \) is the remainder. Given that the remainder is \( (x - 6) \), we can write:
\[ f(x) = (x^2 - 2x - 3) \cdot Q(x) + (x - 6) \]

### Step 2: Use polynomial identities
Since \( g(x) \) is a quadratic polynomial, \( Q(x) \) must be a linear polynomial of the form \( ax + b \). We can express \( f(x) \) as:
\[ f(x) = (x^2 - 2x - 3)(ax + b) + (x - 6) \]

### Step 3: Expand the expression
Now we expand the right-hand side:
\[ (x^2 - 2x - 3)(ax + b) = ax^3 + bx^2 - 2ax^2 - 2bx - 3ax - 3b \]
Combining like terms gives:
\[ = ax^3 + (b - 2a)x^2 + (-2b - 3a)x - 3b \]
Adding the remainder \( (x - 6) \):
\[ f(x) = ax^3 + (b - 2a)x^2 + (-2b - 3a + 1)x + (-3b - 6) \]

### Step 4: Compare coefficients
Now we compare coefficients from \( f(x) = x^3 - 2x^2 + px - q \):
1. Coefficient of \( x^3 \): \( a = 1 \)
2. Coefficient of \( x^2 \): \( b - 2a = -2 \)
3. Coefficient of \( x \): \( -2b - 3a + 1 = p \)
4. Constant term: \( -3b - 6 = -q \)

### Step 5: Solve for \( a \) and \( b \)
From the first equation, we have:
\[ a = 1 \]
Substituting \( a = 1 \) into the second equation:
\[ b - 2(1) = -2 \implies b - 2 = -2 \implies b = 0 \]

### Step 6: Find \( p \) and \( q \)
Substituting \( a = 1 \) and \( b = 0 \) into the third equation:
\[ -2(0) - 3(1) + 1 = p \implies -3 + 1 = p \implies p = -2 \]
Now substituting \( b = 0 \) into the fourth equation:
\[ -3(0) - 6 = -q \implies -6 = -q \implies q = 6 \]

### Final Values
Thus, the values of \( p \) and \( q \) are:
\[ p = -2, \quad q = 6 \]