Question

Perform the operation and simplify completely(1 point) 8x3−12x2+18x−272x−3 −x2+4x =

Answer 1

To simplify the expression \( \frac{8x^3 - 12x^2 + 18x - 27}{2x - 3} - x^2 + 4x \), we can follow these steps:

1. **Divide the polynomial \( 8x^3 - 12x^2 + 18x - 27 \) by \( 2x - 3 \) using polynomial long division.**

2. **Perform the division:**

- Divide the leading term of the numerator \( 8x^3 \) by the leading term of the denominator \( 2x \):
\[
\frac{8x^3}{2x} = 4x^2.
\]
- Multiply \( 4x^2 \) by \( 2x - 3 \) and subtract from the original polynomial:
\[
8x^3 - 12x^2 + 18x - 27 - (4x^2(2x - 3)) = 8x^3 - 12x^2 + 18x - 27 - (8x^3 - 12x^2) = 18x - 27.
\]

- Now, we have \( \frac{18x - 27}{2x - 3} \):
- Divide the leading term \( 18x \) by \( 2x \):
\[
\frac{18x}{2x} = 9.
\]
- 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:
\[
\frac{8x^3 - 12x^2 + 18x - 27}{2x - 3} = 4x^2 + 9.
\]

3. **Now rewrite the expression:**
\[
4x^2 + 9 - x^2 + 4x.
\]

4. **Combine like terms:**
- Combine \( 4x^2 \) and \( -x^2 \):
\[
4x^2 - x^2 = 3x^2.
\]
- The remaining terms are \( 4x + 9 \).

5. **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:

\[
\boxed{3x^2 + 4x + 9}.
\]