If `(2x^3 + 5x^2 - 4x - 6)` is divided by `(2x + 1)`, then the remainder obtained is ?

Options

A. `-13/3`
B. `3`
C. `-3`
D. `6`

Answer

To find the remainder when \( 2x^3 + 5x^2 - 4x - 6 \) is divided by \( 2x + 1 \), we can use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We need to divide \( 2x^3 + 5x^2 - 4x - 6 \) by \( 2x + 1 \).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \( 2x^3 \) by the leading term of the divisor \( 2x \):
\[ \frac{2x^3}{2x} = x^2 \]

### Step 3: Multiply and subtract
Multiply \( x^2 \) by the entire divisor \( 2x + 1 \):
\[ x^2(2x + 1) = 2x^3 + x^2 \]
Now subtract this from the original polynomial:
\[ (2x^3 + 5x^2 - 4x - 6) - (2x^3 + x^2) = (5x^2 - x^2) - 4x - 6 = 4x^2 - 4x - 6 \]

### Step 4: Repeat the process
Now, divide the leading term \( 4x^2 \) by the leading term \( 2x \):
\[ \frac{4x^2}{2x} = 2x \]
Multiply \( 2x \) by the divisor \( 2x + 1 \):
\[ 2x(2x + 1) = 4x^2 + 2x \]
Subtract this from the current polynomial:
\[ (4x^2 - 4x - 6) - (4x^2 + 2x) = (-4x - 2x) - 6 = -6x - 6 \]

### Step 5: Continue dividing
Now, divide the leading term \( -6x \) by the leading term \( 2x \):
\[ \frac{-6x}{2x} = -3 \]
Multiply \( -3 \) by the divisor \( 2x + 1 \):
\[ -3(2x + 1) = -6x - 3 \]
Subtract this from the current polynomial:
\[ (-6x - 6) - (-6x - 3) = -6 + 3 = -3 \]

### Step 6: Conclusion
The remainder when \( 2x^3 + 5x^2 - 4x - 6 \) is divided by \( 2x + 1 \) is:
\[ \text{Remainder} = -3 \]

Thus, the answer is \(-3\).

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