Question

perform the operations and simplify completely 8x^3-12x^2+18x-27/2x-3 -x^2+4x

Answer 1

To simplify the expression \(\frac{8x^3 - 12x^2 + 18x - 27}{2x - 3} - x^2 + 4x\), we will first perform polynomial long division on the first part, \(\frac{8x^3 - 12x^2 + 18x - 27}{2x - 3}\).

### Step 1: Perform Polynomial Long Division

1. **Divide the leading term**: Divide \(8x^3\) by \(2x\) which gives \(4x^2\).
2. **Multiply**: \(4x^2(2x - 3) = 8x^3 - 12x^2\).
3. **Subtract**:
\[
(8x^3 - 12x^2 + 18x - 27) - (8x^3 - 12x^2) = 18x - 27.
\]

4. **Bring down**: Now, use \(2x - 3\) to divide the resulting polynomial \(18x - 27\).
5. **Divide the leading term**: Divide \(18x\) by \(2x\) which gives \(9\).
6. **Multiply**: \(9(2x - 3) = 18x - 27\).
7. **Subtract**:
\[
(18x - 27) - (18x - 27) = 0.
\]

Since the remainder is 0, we conclude that:
\[
\frac{8x^3 - 12x^2 + 18x - 27}{2x - 3} = 4x^2 + 9.
\]

### Step 2: Substitute Back Into the Original Expression
Now we substitute back into the expression:
\[
4x^2 + 9 - x^2 + 4x.
\]

### Step 3: Combine Like Terms
1. **Combine \(4x^2 - x^2\)**:
\[
4x^2 - x^2 = 3x^2.
\]

2. **Bring down the \(4x\)** and the \(9\):
\[
3x^2 + 4x + 9.
\]

### Final Result
Thus, the simplified expression is:
\[
\boxed{3x^2 + 4x + 9}.
\]