If `2x^(3)+3x^(2)+7x+6` be divided by `2x+1`, find the remainder.

Answer

To find the remainder when dividing the polynomial \(2x^3 + 3x^2 + 7x + 6\) by \(2x + 1\), we can use polynomial long division. Here are the steps:

### Step-by-Step Solution:

1. **Set Up the Division**:
We are dividing \(2x^3 + 3x^2 + 7x + 6\) by \(2x + 1\).

2. **Divide the Leading Terms**:
Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(2x\):
\[ \frac{2x^3}{2x} = x^2 \]

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:
\[ (2x^3 + 3x^2 + 7x + 6) - (2x^3 + x^2) = (3x^2 - x^2) + 7x + 6 = 2x^2 + 7x + 6 \]

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

5. **Continue the Division**:
Divide \(6x\) by \(2x\):
\[ \frac{6x}{2x} = 3 \]
Multiply \(3\) by the divisor:
\[ 3(2x + 1) = 6x + 3 \]
Subtract this from \(6x + 6\):
\[ (6x + 6) - (6x + 3) = 6 - 3 = 3 \]

6. **Conclusion**:
The remainder when dividing \(2x^3 + 3x^2 + 7x + 6\) by \(2x + 1\) is \(3\).

### Final Answer:
The remainder is \(3\).

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