To simplify the expression \((8x^3 - 12x^2 + 18x - 27) / (2x - 3)\), we can use polynomial long division.
Step 1: Set up the long division
We want to divide \(8x^3 - 12x^2 + 18x - 27\) by \(2x - 3\).
Step 2: Divide the leading terms
- Divide the leading term of the numerator \(8x^3\) by the leading term of the denominator \(2x\): \[ \frac{8x^3}{2x} = 4x^2 \]
Step 3: Multiply and subtract
- Now multiply \(4x^2\) by the entire divisor \(2x - 3\): \[ 4x^2(2x - 3) = 8x^3 - 12x^2 \]
- Subtract this from the original polynomial: \[ (8x^3 - 12x^2 + 18x - 27) - (8x^3 - 12x^2) = 18x - 27 \]
Step 4: Repeat the process
- Now, divide the new leading term \(18x\) by the leading term of the divisor \(2x\): \[ \frac{18x}{2x} = 9 \]
- Multiply \(9\) by the divisor \(2x - 3\): \[ 9(2x - 3) = 18x - 27 \]
- Subtract this from the current polynomial: \[ (18x - 27) - (18x - 27) = 0 \]
Step 5: Conclusion
Since we reached a remainder of \(0\), the original polynomial \(8x^3 - 12x^2 + 18x - 27\) is exactly divisible by \(2x - 3\).
Thus, the result of the division is: \[ \frac{8x^3 - 12x^2 + 18x - 27}{2x - 3} = 4x^2 + 9 \]
So the simplified expression is: \[ \boxed{4x^2 + 9} \]
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