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The difference of cubes is a special algebraic identity that helps in factoring expressions which are made up of terms like the given one, where two perfect cubes are subtracted from each other.

• It is expressed by the formula: \( A^3 - B^3 = (A - B)(A^2 + AB + B^2) \).
• This formula shows how any expression of the form \( A^3 - B^3 \) can be split into a product of a binomial and a trinomial.

Understanding this identity is helpful for simplifying complex algebraic expressions and solving equations that involve cubes. Being able to recognize when an expression is a difference of cubes allows us to apply this powerful tool to seamlessly factor these expressions.
In this problem, identifying that \( 64a^3 - 125b^6 \) is a difference of cubes is crucial to breaking it down.

Cube roots allow us to express a number or a term that, when raised to the power of three, provides us with the initial expression. Here, "finding the cube root" is a key step in recognizing this form in our problem.**Understanding Cube Roots:**- A cube root \( x \) of a number \( y \) is written as \( x = \sqrt[3]{y} \).- For example, \( \sqrt[3]{64} = 4 \) since \(4^3 = 64\). Similarly, \( \sqrt[3]{125} = 5 \) because \(5^3 = 125\).- To factor a difference of cubes, identifying cube roots accurately is crucial.In the exercise, we determined the cube roots of \( 64a^3 \) and \( 125b^6 \) as \( 4a \) and \( 5b^2 \) respectively, allowing us to apply them in our difference of cubes formula. Recognizing and extracting these roots turns what seems complex into a more manageable algebraic task.

Polynomial factorization is the process of breaking down a polynomial into the product of its smaller components or factors. It's like unearthing the building blocks of an expression.**Steps for Polynomial Factorization in this context:**- **Identify Patterns:** Look for recognizable forms like difference of cubes to determine initial steps.- **Express in Simple Terms:** Use cube roots to simplify parts of the expression, transforming complex terms into manageable components.- **Combine and Simplify:** Once factorized, the terms can be grouped and simplified into a product of binomials or trinomials.In the solution of the problem, once we apply the difference of cubes formula, it involved creating a binomial \((4a - 5b^2)\) and a trinomial \((16a^2 + 20ab^2 + 25b^4)\). Simplifying each part ensured that the original polynomial was neatly factorized into these components.
Mastering polynomial factorization helps in quickly simplifying polynomials, solving equations, and is a foundational skill in algebra.