Question

Perform the operation(s) and simplify completely.

8x3−12x2+18x−272x−3−x2+4x

Answer 1

To perform the operation and simplify the expression \( \frac{8x^3 - 12x^2 + 18x - 27}{2x - 3 - x^2 + 4x} \), we first need to simplify the denominator.

Start by rearranging the denominator terms:
\[
2x - 3 - x^2 + 4x = -x^2 + 2x + 4x - 3 = -x^2 + 6x - 3.
\]
Notice that we can factor out a \(-1\):
\[
-x^2 + 6x - 3 = -(x^2 - 6x + 3).
\]

We now have:
\[
\frac{8x^3 - 12x^2 + 18x - 27}{-(x^2 - 6x + 3)}.
\]

Next, let's perform the polynomial long division of \(8x^3 - 12x^2 + 18x - 27\) by \(- (x^2 - 6x + 3)\).

1. Divide the leading term \(8x^3\) by the leading term of the divisor \(-x^2\):
\[
\frac{8x^3}{-x^2} = -8x.
\]
2. Multiply \(-8x\) by the entire divisor:
\[
-8x(-x^2 + 6x - 3) = 8x^3 - 48x^2 + 24x.
\]
3. Subtract this from the original polynomial:
\[
(8x^3 - 12x^2 + 18x - 27) - (8x^3 - 48x^2 + 24x).
\]
This gives:
\[
(-12x^2 + 48x^2) + (18x - 24x) - 27 = 36x^2 - 6x - 27.
\]
4. Now we need to continue dividing \(36x^2 - 6x - 27\) by \(- (x^2 - 6x + 3)\). Divide:
\[
\frac{36x^2}{-x^2} = -36.
\]
5. Multiply \(-36\) by the entire divisor:
\[
-36(-x^2 + 6x - 3) = 36x^2 - 216x + 108.
\]
6. Subtract this from the remainder:
\[
(36x^2 - 6x - 27) - (36x^2 - 216x + 108).
\]
This gives:
\[
(-6x + 216x) + (-27 - 108) = 210x - 135.
\]

Putting it all together, we have:
\[
-8x - 36 + \frac{210x - 135}{-(x^2 - 6x + 3)}.
\]
This can be simplified further by factoring or simplifying the remainder term if possible. But overall the expression is simplified to:
\[
-8x - 36 - \frac{210x - 135}{x^2 - 6x + 3}.
\]

You can factor \(210x - 135\):
\[
= 15(14x - 9).
\]

Final answer:
\[
-8x - 36 - \frac{15(14x - 9)}{-(x^2 - 6x + 3)}.
\]