On dividing the polynomial `8x^(3)-6x^(2) + 10x + 3` by `(4x+1)`, the quotient is `2x^(2) +k`, where, k is equal to

Options

A. `3-2x`
B. `-3+2x`
C. `3+2x`
D. `-3-2x`

Answer

To find the value of \( k \) in the polynomial division of \( 8x^3 - 6x^2 + 10x + 3 \) by \( 4x + 1 \) where the quotient is given as \( 2x^2 + k \), we can follow these steps:

### Step 1: Set up the polynomial division
We need to divide \( 8x^3 - 6x^2 + 10x + 3 \) by \( 4x + 1 \).

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

**Hint:** Always start polynomial long division by dividing the leading terms.

### Step 3: Multiply and subtract
Now, multiply \( 2x^2 \) by the entire divisor \( 4x + 1 \):
\[ 2x^2 \cdot (4x + 1) = 8x^3 + 2x^2 \]
Subtract this from the original polynomial:
\[ (8x^3 - 6x^2 + 10x + 3) - (8x^3 + 2x^2) = -8x^2 + 10x + 3 \]

**Hint:** When subtracting, change the signs of the terms you are subtracting.

### Step 4: Repeat the process
Now, take the new leading term \(-8x^2\) and divide it by \(4x\):
\[ \frac{-8x^2}{4x} = -2x \]
This gives us the next term of the quotient.

**Hint:** Continue the process of dividing the leading terms until you reach a term that cannot be divided anymore.

### Step 5: Multiply and subtract again
Multiply \(-2x\) by the divisor \(4x + 1\):
\[ -2x \cdot (4x + 1) = -8x^2 - 2x \]
Subtract this from \(-8x^2 + 10x + 3\):
\[ (-8x^2 + 10x + 3) - (-8x^2 - 2x) = 12x + 3 \]

**Hint:** Keep track of the signs carefully when performing subtraction.

### Step 6: Divide the last term
Now, take \(12x\) and divide it by \(4x\):
\[ \frac{12x}{4x} = 3 \]
This gives us the last term of the quotient.

**Hint:** The last term will be the constant term in the quotient.

### Step 7: Final multiplication and subtraction
Multiply \(3\) by the divisor \(4x + 1\):
\[ 3 \cdot (4x + 1) = 12x + 3 \]
Subtract this from \(12x + 3\):
\[ (12x + 3) - (12x + 3) = 0 \]

**Hint:** If you reach zero, it means the division is complete with no remainder.

### Step 8: Write the complete quotient
The complete quotient from the division is:
\[ 2x^2 - 2x + 3 \]
We know from the problem statement that this is equal to \(2x^2 + k\).

### Step 9: Solve for \( k \)
Now, equate the two expressions:
\[ 2x^2 - 2x + 3 = 2x^2 + k \]
By comparing the constant terms, we find:
\[ k = -2x + 3 \]

Since \( k \) is a constant, we can conclude:
\[ k = 3 \quad \text{(when } x = 0\text{)} \]

### Final Answer:
Thus, the value of \( k \) is \( 3 \).

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