To perform long division with the polynomial \(3x^3 - 9x^2 - 72\) divided by \(3x^2 - 2\), follow these steps:
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Set up the division: Write \(3x^3 - 9x^2 - 72\) under the long division symbol and \(3x^2 - 2\) outside.
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Divide the leading terms: Divide the leading term of the numerator by the leading term of the denominator: \[ \frac{3x^3}{3x^2} = x \] The first term of the quotient is \(x\).
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Multiply and subtract: Multiply the entire divisor \(3x^2 - 2\) by \(x\) and subtract from the original polynomial: \[ x(3x^2 - 2) = 3x^3 - 2x \] Now, subtract this from the original polynomial: \[ (3x^3 - 9x^2 - 72) - (3x^3 - 2x) = -9x^2 + 2x - 72 \]
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Bring down the next term (if needed): There are no additional terms to bring down because we are still working with the remainder \(-9x^2 + 2x - 72\).
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Repeat the process: Divide the leading term of the new polynomial by the leading term of the divisor: \[ \frac{-9x^2}{3x^2} = -3 \] Add this to the quotient: \(x - 3\).
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Multiply and subtract again: Multiply the entire divisor \(3x^2 - 2\) by \(-3\): \[ -3(3x^2 - 2) = -9x^2 + 6 \] Subtract this from the previous result: \[ (-9x^2 + 2x - 72) - (-9x^2 + 6) = 2x - 66 \]
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Final remainder: Now we continue with \(2x - 66\). The division is complete, but you can still divide: \[ \frac{2x}{3x^2 - 2} \] This gives you the remainder.
So, the result of the division is: \[ \text{Quotient: } x - 3 \quad \text{Remainder: } \frac{2x - 66}{3x^2 - 2} \]
Alternatively, you can write the full expression as: \[ x - 3 + \frac{2x - 66}{3x^2 - 2} \]
Putting it all together, the final answer is: \[ \boxed{x - 3 + \frac{2x - 66}{3x^2 - 2}} \]