Problem 79 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 64 x^{3}-y^{6} $$

Step-by-Step Solution

Answer

\((4x - y^2)(16x^2 + 4xy^2 + y^4)\)

Step 1: Recognize the Polynomial Form

Notice that the polynomial given is \(64x^3 - y^6\). This expression contains two terms: \(64x^3\) and \(y^6\), and it can be recognized as a difference of two cubes of the form \(a^3 - b^3\).

Step 2: Express Each Term as a Cube

Write \(64x^3\) as \((4x)^3\) and \(y^6\) as \((y^2)^3\). Thus, the expression becomes \((4x)^3 - (y^2)^3\).

Step 3: Apply the Difference of Cubes Formula

Use the formula for factoring a difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). In this case, let \(a = 4x\) and \(b = y^2\). Substituting into the formula gives: \((4x - y^2)((4x)^2 + 4x(y^2) + (y^2)^2)\).

Step 4: Simplify the Resulting Expression

Simplifying the expression, you get \(4x - y^2\) for the first factor. For the second factor, calculate: \((4x)^2 = 16x^2\), \(4x(y^2) = 4xy^2\), and \((y^2)^2 = y^4\). Thus, the second factor becomes \(16x^2 + 4xy^2 + y^4\).

Step 5: Write the Final Factored Form

The fully factored form of the given polynomial is \((4x - y^2)(16x^2 + 4xy^2 + y^4)\).

Key Concepts

Difference of Cubes

In mathematics, the difference of cubes is a special pattern used in polynomial factoring. It applies when you have an expression of the form \(a^3 - b^3\). Recognizing this pattern allows you to simplify the expression using a specific formula:

• \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

The first expression, \(a - b\), is straightforward, while the second expression, \(a^2 + ab + b^2\), combines the squares and products of the variables. Understanding the difference of cubes helps in breaking down complex algebraic expressions into their simpler components. This makes it easier to solve equations and other mathematical problems. In our exercise, recognizing \(64x^3 - y^6\) as a difference of cubes was a crucial step in the factoring process.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a specific value or set of values. These expressions do not need to have an equality sign, unlike equations.
They can include constants (like 64), variables (like \(x\) and \(y\)), coefficients (the numbers in front of the variables), and exponents (such as 3 or 6). In the exercise, \(64x^3 - y^6\) is an algebraic expression that students need to factor.
Comprehending algebraic expressions involves understanding how the components interact, which is vital in simplifying and manipulating them. This skill is essential in algebra as well as in more advanced math topics. Remember, the better you understand how algebraic expressions work, the easier it becomes to apply different techniques for their simplification or transformation.

Factoring Techniques

Factoring is a fundamental algebraic process involving the breaking down of expressions into simpler components called factors. These techniques are crucial for simplifying expressions, solving equations, and understanding mathematical relationships. There are several methods of factoring, including:

• Factoring by grouping
• Using special formulas (like the difference of squares or cubes)
• Trial and error

In the exercise, the difference of cubes technique was used to factor the polynomial \(64x^3 - y^6\). Each term was rewritten as a cube—\((4x)^3 - (y^2)^3\)—which facilitated applying the specific formula for difference of cubes. By using this technique, complex expressions become clearer and more manageable, assisting in further problem-solving endeavors.

Mathematics Education

Mathematics education focuses on developing students' understanding and skills in various math concepts and techniques. Factoring polynomials, such as with the difference of cubes, is a key subject taught to deepen students' comprehension of algebra. By learning how to identify special patterns and formulas, students improve their algebraic problem-solving capabilities.
Effective mathematics education employs clear explanations, step-by-step instructions, and numerous examples to build a solid foundation. As shown in the original solution, each step was broken down—from recognizing the expression form to simplifying the factored form—enabling students to grasp the concept thoroughly. When students understand how to use these techniques, they boost their confidence and ability to tackle more intricate mathematical problems, creating a strong base for advanced studies.