To divide the polynomial \(3x^3 + x^2 - 10x - 9\) by \(x + 1\) using long division, we will follow these steps:
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Write the dividend \(3x^3 + x^2 - 10x - 9\) under the long division symbol and the divisor \(x + 1\) outside.
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Divide the leading term of the dividend \(3x^3\) by the leading term of the divisor \(x\) to get \(3x^2\).
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Multiply \(3x^2\) by \(x + 1\) to get \(3x^3 + 3x^2\).
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Subtract \(3x^3 + 3x^2\) from the dividend: \[ (3x^3 + x^2) - (3x^3 + 3x^2) = -2x^2 \]
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Bring down the \(-10x\) to get \(-2x^2 - 10x\).
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Divide the leading term \(-2x^2\) by \(x\) to get \(-2x\).
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Multiply \(-2x\) by \(x + 1\) to get \(-2x^2 - 2x\).
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Subtract \(-2x^2 - 2x\) from \(-2x^2 - 10x\): \[ (-2x^2 - 10x) - (-2x^2 - 2x) = -8x \]
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Bring down the \(-9\) to get \(-8x - 9\).
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Divide \(-8x\) by \(x\) to get \(-8\).
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Multiply \(-8\) by \(x + 1\) to get \(-8x - 8\).
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Subtract \(-8x - 8\) from \(-8x - 9\): \[ (-8x - 9) - (-8x - 8) = -1 \]
The result of the division is: \[ 3x^2 - 2x - 8 \] With a remainder of \(-1\).
Putting this into the required form \( q(x) + \frac{r(x)}{b(x)} \), we have: \[ \frac{3x^3 + x^2 - 10x - 9}{x + 1} = 3x^2 - 2x - 8 - \frac{1}{x + 1} \]
So the answer is: \[ q(x) = 3x^2 - 2x - 8, \quad r(x) = -1, \quad b(x) = x + 1 \] Final expression: \[ 3x^2 - 2x - 8 - \frac{1}{x + 1} \]
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