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The 'sum of cubes' is a special algebraic expression that represents the sum of two cubed terms. In algebra, recognizing such patterns can simplify complex expressions. For instance, in our exercise, we saw the binomial 125x\textsuperscript{3} + 8. Here, 125x\textsuperscript{3} can be written as \( (5x)^3 \) and 8 can be written as \( 2^3 \). This recognition is crucial to applying the sum of cubes formula.
The sum of cubes formula states:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
By identifying a = 5x and b = 2, we are able to substitute these values back into the formula for easier computation.

Polynomial factorization is an essential skill in algebra, helping break down polynomials into simpler factors. When factoring polynomials, it's useful to identify special patterns like the sum of cubes or difference of squares. In our case, the binomial 125x\textsuperscript{3} + 8 is factored by recognizing it as the sum of cubes.
Once identified, apply the sum of cubes formula:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
The first step involves substituting the identified parts, a = 5x and b = 2. This transforms the expression into (5x + 2).The second factor is a bit more complex, as we calculate each term:
- \( (5x)^2 = 25x^2 \)
- \( - (5x)(2) = -10x \)
- \( (2)^2 = 4 \)
So, combining everything, it becomes:\( (5x + 2)(25x^2 - 10x + 4) \)
This makes the original expression much simpler.

Algebraic formulas are tools that make solving equations and factoring expressions manageable. They save time and simplify calculations. In this exercise, we used the sum of cubes formula:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Such formulas are derived from algebraic identities, which are always true for any variables they contain.
Understanding how to use these formulas is vital in mastering algebra.
Let's break down its application:
- Identify if the expression fits the formula by recognizing cubes.
- Substitute the identified parts into the formula.
- Expand and simplify, if necessary.
Algebraic formulas are powerful because they take abstract concepts and make them easy to use with concrete steps, perfect for solving problems like our given binomial.