To perform polynomial long division for the expression \((6x^4 - 11x^3 - 7x^2 + 8x + 3) \div (3x + 2)\), follow these steps:
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Divide the leading term of the dividend \(6x^4\) by the leading term of the divisor \(3x\): \[ \frac{6x^4}{3x} = 2x^3. \]
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Multiply \(2x^3\) by the entire divisor \(3x + 2\): \[ 2x^3 \cdot (3x + 2) = 6x^4 + 4x^3. \]
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Subtract this result from the original polynomial: \[ (6x^4 - 11x^3 - 7x^2 + 8x + 3) - (6x^4 + 4x^3) = -15x^3 - 7x^2 + 8x + 3. \]
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Bring down the next term and repeat the process. Divide \(-15x^3\) by \(3x\): \[ \frac{-15x^3}{3x} = -5x^2. \]
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Multiply \(-5x^2\) by \(3x + 2\): \[ -5x^2 \cdot (3x + 2) = -15x^3 - 10x^2. \]
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Subtract this from the current polynomial: \[ (-15x^3 - 7x^2 + 8x + 3) - (-15x^3 - 10x^2) = 3x^2 + 8x + 3. \]
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Now divide \(3x^2\) by \(3x\): \[ \frac{3x^2}{3x} = x. \]
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Multiply \(x\) by \(3x + 2\): \[ x \cdot (3x + 2) = 3x^2 + 2x. \]
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Subtract: \[ (3x^2 + 8x + 3) - (3x^2 + 2x) = 6x + 3. \]
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Now divide \(6x\) by \(3x\): \[ \frac{6x}{3x} = 2. \]
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Multiply: \[ 2 \cdot (3x + 2) = 6x + 4. \]
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Subtract: \[ (6x + 3) - (6x + 4) = -1. \]
So the final result of the division is: \[ 2x^3 - 5x^2 + x + 2 - \frac{1}{3x + 2}. \]
In this context, the correct answer using the choices provided would be: \[ \text{Option B: } 2x^3 - 5x^2 + x + 2 - \frac{1}{3x + 2}. \]