To simplify the expression \( \frac{8x^3 - 12x^2 + 18x - 27}{2x - 3} - x^2 + 4x \), we can follow these steps:
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Divide the polynomial \( 8x^3 - 12x^2 + 18x - 27 \) by \( 2x - 3 \) using polynomial long division.
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Perform the division:
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Divide the leading term of the numerator \( 8x^3 \) by the leading term of the denominator \( 2x \): \[ \frac{8x^3}{2x} = 4x^2. \]
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Multiply \( 4x^2 \) by \( 2x - 3 \) and subtract from the original polynomial: \[ 8x^3 - 12x^2 + 18x - 27 - (4x^2(2x - 3)) = 8x^3 - 12x^2 + 18x - 27 - (8x^3 - 12x^2) = 18x - 27. \]
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Now, we have \( \frac{18x - 27}{2x - 3} \):
- Divide the leading term \( 18x \) by \( 2x \): \[ \frac{18x}{2x} = 9. \]
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Multiply \( 9 \) by \( 2x - 3 \) and subtract: \[ 18x - 27 - (9(2x - 3)) = 18x - 27 - (18x - 27) = 0. \]
Therefore, the result of the polynomial long division is: \[ \frac{8x^3 - 12x^2 + 18x - 27}{2x - 3} = 4x^2 + 9. \]
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Now rewrite the expression: \[ 4x^2 + 9 - x^2 + 4x. \]
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Combine like terms:
- Combine \( 4x^2 \) and \( -x^2 \): \[ 4x^2 - x^2 = 3x^2. \]
- The remaining terms are \( 4x + 9 \).
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Final result: \[ 3x^2 + 4x + 9. \]
Thus, the simplified form of the expression \( \frac{8x^3 - 12x^2 + 18x - 27}{2x - 3} - x^2 + 4x \) is:
\[ \boxed{3x^2 + 4x + 9}. \]
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