Question

Perform the operation(s) and simplify completely.8x3−12x2+18x−272x−3−x2+4x

Answer 1

To simplify the expression \( \frac{8x^3 - 12x^2 + 18x - 27}{2x - 3 - x^2 + 4x} \), first, let's simplify the denominator.

First, combine like terms in the denominator:
\[
2x - 3 - x^2 + 4x = 6x - x^2 - 3 = -x^2 + 6x - 3
\]
Thus, the expression can be rewritten as:
\[
\frac{8x^3 - 12x^2 + 18x - 27}{-x^2 + 6x - 3}
\]

Now to simplify this, we need to perform polynomial long division or factor if possible.

1. **Factor the denominator** \(-x^2 + 6x - 3\):
We can factor this by finding the roots using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-6 \pm \sqrt{6^2 - 4 \cdot (-1) \cdot (-3)}}{2 \cdot (-1)} = \frac{-6 \pm \sqrt{36 - 12}}{-2} = \frac{-6 \pm \sqrt{24}}{-2}
\]
\[
= \frac{-6 \pm 2\sqrt{6}}{-2} = 3 \mp \sqrt{6}
\]

Thus, the roots are \(3 - \sqrt{6}\) and \(3 + \sqrt{6}\). The factorization of the quadratic is:
\[
-1 \cdot (x - (3 - \sqrt{6}))(x - (3 + \sqrt{6}))
\]
or simply:
\[
-(x^2 - 6x + 3)
\]

2. **Perform polynomial long division** for the numerator \(8x^3 - 12x^2 + 18x - 27\) with the factored form of the denominator (or just the quadratic, \( -x^2 + 6x - 3\)):
We'll divide the leading term:
\[
\frac{8x^3}{-x^2} = -8x
\]

Multiply \( -8x \) by the entire denominator:
\[
-8x(-x^2 + 6x - 3) = 8x^3 - 48x^2 + 24x
\]

Subtract this from the numerator:
\[
(8x^3 - 12x^2 + 18x - 27) - (8x^3 - 48x^2 + 24x) = ( -12x^2 + 48x^2) + (18x - 24x) - 27 = 36x^2 - 6x - 27
\]

Now we divide \( 36x^2 - 6x - 27 \) by \( -x^2 + 6x - 3 \).

Next, repeat the process:
Divide \( 36x^2 \) by \( -x^2 \) to get \( -36 \).
\[
-36(-x^2 + 6x - 3) = 36x^2 - 216x + 108
\]
Subtract this:
\[
(36x^2 - 6x - 27) - (36x^2 - 216x + 108) = - 6x + 216x - 27 - 108
= 210x - 135
\]

Nice, our remainder is \( 210x - 135 \).

Thus, putting it all together:
\[
\frac{8x^3 - 12x^2 + 18x - 27}{-(x^2 - 6x + 3)} = -8x - 36 + \frac{210x - 135}{-(x^2 - 6x + 3)}
\]
We can leave it in this form or further factor the remainder \( 210x - 135 \).

Final result (depending on how simplified you want it):
\[
-8x - 36 + \frac{135 - 210x}{x^2 - 6x + 3}
\]

This is the simplified or factored form depending on further instructions required.