To perform long division for the polynomial \(3x^3 + 2x^2 - 18x + 9\) divided by \(x + 3\), we follow these steps:
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Set up the long division: We write \(3x^3 + 2x^2 - 18x + 9\) under the long division symbol and \(x + 3\) outside.
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Divide the leading terms: Divide the leading term of the dividend (\(3x^3\)) by the leading term of the divisor (\(x\)): \[ \frac{3x^3}{x} = 3x^2 \] This is the first term of the quotient.
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Multiply and subtract: Multiply the entire divisor \(x + 3\) by \(3x^2\): \[ 3x^2(x + 3) = 3x^3 + 9x^2 \] Now subtract this from the original polynomial: \[ (3x^3 + 2x^2 - 18x + 9) - (3x^3 + 9x^2) = 2x^2 - 9x^2 - 18x + 9 = -7x^2 - 18x + 9 \]
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Repeat the process: Divide the new leading term (\(-7x^2\)) by the leading term of the divisor (\(x\)): \[ \frac{-7x^2}{x} = -7x \] Multiply the divisor by \(-7x\): \[ -7x(x + 3) = -7x^2 - 21x \] Subtract again: \[ (-7x^2 - 18x + 9) - (-7x^2 - 21x) = -18x + 21x + 9 = 3x + 9 \]
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Repeat again: Divide \(3x\) by \(x\): \[ \frac{3x}{x} = 3 \] Multiply the divisor by \(3\): \[ 3(x + 3) = 3x + 9 \] Subtract: \[ (3x + 9) - (3x + 9) = 0 \]
At this point, there are no more terms to bring down, and the remainder is zero.
- Final quotient: Thus, the result of the long division is: \[ \boxed{3x^2 - 7x + 3} \] The polynomial \(3x^3 + 2x^2 - 18x + 9\) divided by \(x + 3\) gives a quotient of \(3x^2 - 7x + 3\) with a remainder of 0.
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