The quotient obtained on dividing `(8x^(4)-2x^(2)+6x-7)` by `(2x+1)` is `(4x^(3)+px^(2)-qx+3)` . The value of (q-p) is

Options

A. `0`
B. `-2`
C. `2`
D. `4`

Answer

To solve the problem, we need to perform polynomial long division of \(8x^4 - 2x^2 + 6x - 7\) by \(2x + 1\) and then compare the result with the given quotient \(4x^3 + px^2 - qx + 3\).

### Step-by-Step Solution:

1. **Set up the division**: We are dividing \(8x^4 - 2x^2 + 6x - 7\) by \(2x + 1\).

2. **Divide the leading terms**:
- Divide the leading term of the dividend \(8x^4\) by the leading term of the divisor \(2x\):
\[ \frac{8x^4}{2x} = 4x^3 \]
This gives us the first term of the quotient.

3. **Multiply and subtract**:
- Multiply \(4x^3\) by \(2x + 1\):
\[ 4x^3 \cdot (2x + 1) = 8x^4 + 4x^3 \]
- Subtract this from the original polynomial:
\[ (8x^4 - 2x^2 + 6x - 7) - (8x^4 + 4x^3) = -4x^3 - 2x^2 + 6x - 7 \]

4. **Repeat the process**:
- Now, divide the leading term \(-4x^3\) by \(2x\):
\[ \frac{-4x^3}{2x} = -2x^2 \]
- Multiply \(-2x^2\) by \(2x + 1\):
\[ -2x^2 \cdot (2x + 1) = -4x^3 - 2x^2 \]
- Subtract:
\[ (-4x^3 - 2x^2 + 6x - 7) - (-4x^3 - 2x^2) = 6x - 7 \]

5. **Continue dividing**:
- Divide \(6x\) by \(2x\):
\[ \frac{6x}{2x} = 3 \]
- Multiply \(3\) by \(2x + 1\):
\[ 3 \cdot (2x + 1) = 6x + 3 \]
- Subtract:
\[ (6x - 7) - (6x + 3) = -10 \]

6. **Final result**:
- The remainder is \(-10\), and the complete quotient is:
\[ 4x^3 - 2x^2 + 3 \]

7. **Compare with the given quotient**:
- We have \(4x^3 - 2x^2 + 0x + 3\) and the given quotient is \(4x^3 + px^2 - qx + 3\).
- From this, we can see:
- \(p = -2\)
- \(q = 0\)

8. **Calculate \(q - p\)**:
\[ q - p = 0 - (-2) = 2 \]

### Final Answer:
The value of \(q - p\) is \(2\).