Question

Perform the operations and simplify completely (8x^3-12x^2+18x-27)/2x-3

Answer 1

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:
\[
\boxed{4x^2 + 9}
\]