If the polynomial `8x^(4)+14x^(3)-2x^(2)+px+q` is exactly divisible by `4x^(2)+3x-2`, then the values of p and q respectively are

Options

A. 2 and 0
B. `-7` and 2
C. 5 and `-3`
D. 4 and `-1`

Answer

To find the values of \( p \) and \( q \) such that the polynomial \( 8x^4 + 14x^3 - 2x^2 + px + q \) is exactly divisible by \( 4x^2 + 3x - 2 \), we will perform polynomial long division and set the remainder equal to zero.

### Step-by-Step Solution:

1. **Set Up the Division**:
We need to divide \( 8x^4 + 14x^3 - 2x^2 + px + q \) by \( 4x^2 + 3x - 2 \).

2. **Divide the Leading Terms**:
Divide the leading term of the dividend \( 8x^4 \) by the leading term of the divisor \( 4x^2 \):
\[ \frac{8x^4}{4x^2} = 2x^2 \]

3. **Multiply and Subtract**:
Multiply \( 2x^2 \) by the entire divisor \( 4x^2 + 3x - 2 \):
\[ 2x^2 \cdot (4x^2 + 3x - 2) = 8x^4 + 6x^3 - 4x^2 \]
Now subtract this from the original polynomial:
\[ (8x^4 + 14x^3 - 2x^2 + px + q) - (8x^4 + 6x^3 - 4x^2) = (14x^3 - 6x^3) + (-2x^2 + 4x^2) + px + q \]
This simplifies to:
\[ 8x^3 + 2x^2 + px + q \]

4. **Repeat the Process**:
Now, divide the leading term \( 8x^3 \) by \( 4x^2 \):
\[ \frac{8x^3}{4x^2} = 2x \]
Multiply \( 2x \) by the divisor:
\[ 2x \cdot (4x^2 + 3x - 2) = 8x^3 + 6x^2 - 4x \]
Subtract this from the current polynomial:
\[ (8x^3 + 2x^2 + px + q) - (8x^3 + 6x^2 - 4x) = (2x^2 - 6x^2) + (px + 4x) + q \]
This simplifies to:
\[ -4x^2 + (p + 4)x + q \]

5. **Final Division**:
Divide the leading term \( -4x^2 \) by \( 4x^2 \):
\[ \frac{-4x^2}{4x^2} = -1 \]
Multiply \( -1 \) by the divisor:
\[ -1 \cdot (4x^2 + 3x - 2) = -4x^2 - 3x + 2 \]
Subtract this from the current polynomial:
\[ (-4x^2 + (p + 4)x + q) - (-4x^2 - 3x + 2) = ((p + 4) + 3)x + (q - 2) \]
This simplifies to:
\[ (p + 7)x + (q - 2) \]

6. **Set the Remainder to Zero**:
Since the polynomial is exactly divisible, the remainder must be zero:
\[ p + 7 = 0 \quad \text{and} \quad q - 2 = 0 \]
Solving these equations gives:
\[ p = -7 \quad \text{and} \quad q = 2 \]

### Final Answer:
The values of \( p \) and \( q \) are:
\[ p = -7, \quad q = 2 \]