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The sum of cubes is a special algebraic identity that helps to factor expressions involving the sum of two cubes. For example, in the expression \( x^3 + \frac{1}{27} \), we recognize it as a sum of cubes because \(\frac{1}{27}\) can be rewritten as \(\frac{1}{3^3}\) or \( \frac{1}{3}^3 \). To factor a sum of cubes, we use the formula:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \].
In our given expression, \(a\) is \(x\) and \(b\) is \( \frac{1}{3} \). By identifying these parts, we can break down and rewrite the sum of cubes into a product of two simpler expressions.

Algebraic factoring is the process of expressing a polynomial as a product of its factors. In the given problem \( x^3 + \frac{1}{27} \), after recognizing it as a sum of cubes, we use the sum of cubes formula to factor it.
The formula \( (a + b)(a^2 - ab + b^2) \) lets us rewrite \( x^3 + \frac{1}{27} \) as: \( (x + \frac{1}{3})(x^2 - x \frac{1}{3} + (\frac{1}{3})^2) \).
Here, \( x \) and \( \frac{1}{3} \) are the roots which make it easier to handle and simplify the original expression. Factoring helps in reducing expressions to simpler forms, and in solving polynomial equations by setting each factor to zero.

Once we have factored the expression using the sum of cubes formula, we need to simplify it further. Simplifying involves carrying out arithmetic operations inside the parentheses. For our problem:
\( (x + \frac{1}{3})(x^2 - \frac{x}{3} + \frac{1}{9}) \),
we distribute and combine like terms:
- \( x + \frac{1}{3} \) is already simplified.
- In the term \( x^2 - \frac{x}{3} + \frac{1}{9}\), ensure that each part is simplified: \( x^2 \), \( \frac{-x}{3} \), and \( \frac{1}{9} \).
Simplifying these individual parts results in:
\( x^2 - \frac{x}{3} + \frac{1}{9} \) which cannot be simplified further.
Therefore, the final solution remains factored and simplified as:
\( (x + \frac{1}{3})(x^2 - \frac{x}{3} + \frac{1}{9}) \). This step-by-step process ensures we have fully factored and simplified the given polynomial expression.