Question

Simplify the following expression:
[(6x^4 -9x^3 +12x^2 -18x)/(3x-2)]-x^3 +5

Answer 1

To simplify the expression

\[
\frac{6x^4 - 9x^3 + 12x^2 - 18x}{3x - 2} - x^3 + 5,
\]

we start by simplifying the fraction \(\frac{6x^4 - 9x^3 + 12x^2 - 18x}{3x - 2}\).

1. **Factor the numerator**: Factoring out common terms in the numerator:

\[
6x^4 - 9x^3 + 12x^2 - 18x = 3x(2x^3 - 3x^2 + 4x - 6).
\]

Next, we will perform polynomial long division to divide \(2x^3 - 3x^2 + 4x - 6\) by \(3x - 2\).

2. **Polynomial Long Division**:
- Divide \(2x^3\) by \(3x\), which gives \(\frac{2}{3}x^2\).
- Multiply \(\frac{2}{3}x^2\) by \(3x - 2\):
\[
\frac{2}{3}x^2 (3x - 2) = 2x^3 - \frac{4}{3}x^2.
\]
- Subtract this from \(2x^3 - 3x^2 + 4x - 6\):
\[
\left(2x^3 - 3x^2 + 4x - 6\right) - \left(2x^3 - \frac{4}{3}x^2\right) = -3x^2 + \frac{4}{3}x^2 + 4x - 6 = -\frac{5}{3}x^2 + 4x - 6.
\]

- Now, divide \(-\frac{5}{3}x^2\) by \(3x\):
\[
-\frac{5}{9}x.
\]
- Multiply and subtract:
\[
-\frac{5}{9}x(3x - 2) = -\frac{5}{3}x^2 + \frac{10}{9}x.
\]
Resulting in:
\[
\left(-\frac{5}{3}x^2 + 4x - 6\right) - \left(-\frac{5}{3}x^2 + \frac{10}{9}x\right) = \left(4 - \frac{10}{9}\right)x - 6 = \frac{26}{9}x - 6.
\]

- Divide \(\frac{26}{9}x\) by \(3x\):
\[
\frac{26}{27}.
\]
- Multiply:
\[
\frac{26}{27}(3x - 2) = \frac{26}{9}x - \frac{52}{27}.
\]
- Subtract:
\[
\left(\frac{26}{9}x - 6\right) - \left(\frac{26}{9}x - \frac{52}{27}\right) = -6 + \frac{52}{27} = -\frac{162}{27} + \frac{52}{27} = -\frac{110}{27}.
\]

So, we can write:

\[
\frac{2x^3 - 3x^2 + 4x - 6}{3x - 2} = \frac{2}{3}x^2 - \frac{5}{9}x + \frac{26}{27} - \frac{110}{27(3x-2)}.
\]

Putting this back into the expression:

\[
3x\left(\frac{2}{3}x^2 - \frac{5}{9}x + \frac{26}{27}\right) - x^3 + 5.
\]

3. **Combine like terms**:

This leads to:

\[
2x^3 - \frac{5}{3}x^2 + \frac{26}{9} - x^3 + 5,
\]

or

\[
(2x^3 - x^3) - \frac{5}{3}x^2 + 5 + \frac{26}{9}.
\]

This simplifies to:

\[
x^3 - \frac{5}{3}x^2 + \frac{26 + 45}{9} = x^3 - \frac{5}{3}x^2 + \frac{71}{9}.
\]

Thus, the final simplified expression is

\[
\boxed{x^3 - \frac{5}{3}x^2 + \frac{71}{9}}.
\]