Divide. $$ \frac{27 x^{6}-6 x^{3}}{3 x^{3}} $$ Simplify your answer as much as possible.

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Alright, I need to simplify the expression $$\frac{27x^{6} - 6x^{3}}{3x^{3}}$$. Let's break this down step by step to ensure I cover all possible angles and don't miss anything.

First, I observe that the numerator is a difference of two terms: $$27x^{6}$$ and $$6x^{3}$$. The denominator is $$3x^{3}$$. My goal is to simplify this fraction as much as possible.

One approach is to factor out common terms in the numerator before performing the division. Let's see if that's feasible.

Looking at $$27x^{6}$$ and $$6x^{3}$$, I notice that both terms have an $$x^{3}$$ component. Specifically, $$27x^{6}$$ can be written as $$27x^{3} \times x^{3}$$, and $$6x^{3}$$ is already in a form that shows the $$x^{3}$$ factor.

So, factoring out $$x^{3}$$ from the numerator gives:
$$
27x^{6} - 6x^{3} = x^{3}(27x^{3} - 6)
$$
Now, the expression becomes:
$$
\frac{x^{3}(27x^{3} - 6)}{3x^{3}}
$$
At this point, I can see that $$x^{3}$$ appears in both the numerator and the denominator. Since division by $$x^{3}$$ cancels out the $$x^{3}$$ in the numerator, I'm left with:
$$
\frac{27x^{3} - 6}{3}
$$
Now, I can simplify this further by dividing each term in the numerator by 3:
$$
\frac{27x^{3}}{3} - \frac{6}{3} = 9x^{3} - 2
$$
So, the simplified form of the original expression is $$9x^{3} - 2$$.

But wait, let me double-check to ensure I didn't make any mistakes. Starting from the beginning:
$$
\frac{27x^{6} - 6x^{3}}{3x^{3}} = \frac{27x^{3} \times x^{3} - 6x^{3}}{3x^{3}} = \frac{x^{3}(27x^{3} - 6)}{3x^{3}} = \frac{27x^{3} - 6}{3} = 9x^{3} - 2
$$
Yes, that seems consistent. Alternatively, I could have approached this by dividing each term in the numerator by the denominator separately:
$$
\frac{27x^{6}}{3x^{3}} - \frac{6x^{3}}{3x^{3}} = 9x^{3} - 2
$$
Both methods lead to the same result, which reinforces that $$9x^{3} - 2$$ is indeed the correct simplified form.

Another way to verify is by plugging in a value for $$x$$ and checking if both the original and simplified expressions yield the same result. Let's choose $$x = 1$$:
$$
\text{Original: } \frac{27(1)^{6} - 6(1)^{3}}{3(1)^{3}} = \frac{27 - 6}{3} = \frac{21}{3} = 7
$$
$$
\text{Simplified: } 9(1)^{3} - 2 = 9 - 2 = 7
$$
Both expressions equal 7 when $$x = 1$$, which is a good consistency check.

Additionally, considering the degrees of the terms, the original expression has a degree of 6 in the numerator and 3 in the denominator, resulting in a degree of 3 in the simplified expression, which makes sense.

In conclusion, after multiple verification methods, the simplified form of the given expression is confidently $$9x^{3} - 2$$.

$$9x^{3} - 2$$

Divide. \[ \frac{27 x^{6}-6 x^{3}}{3 x^{3}} \] Simplify your answer as much as possible.

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