Problem 61 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ x^{6}-8 y^{3} $$

Step-by-Step Solution

Answer

The completely factored form is \((x^2 - 2y)(x^4 + 2x^2y + 4y^2)\).

Step 1: Identify a Difference of Cubes

Notice the expression \(x^6 - 8y^3\) seems to fit the pattern for a difference of cubes formula: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Here, \(x^6 = (x^2)^3\) and \(8y^3 = (2y)^3\). So the expression is indeed a difference of cubes.

Step 2: Apply the Difference of Cubes Formula

Using the formula for a difference of cubes: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\), set \(a = x^2\) and \(b = 2y\). Substitute these into the formula to get: \((x^2 - 2y)((x^2)^2 + x^2(2y) + (2y)^2)\).

Step 3: Simplify the Expression

Expand the terms within the factors: \((x^2)^2 = x^4\) \(x^2(2y) = 2x^2y\) \((2y)^2 = 4y^2\) Thus, the expression becomes: \((x^2 - 2y)(x^4 + 2x^2y + 4y^2)\).

Step 4: Verify the Complete Factorization

Check if any further factorization is possible. The second factor \(x^4 + 2x^2y + 4y^2\) does not factor further since it has no further common factors or follows a recognizable factorization pattern (like a perfect square trinomial). Therefore, the expression is fully factored as \((x^2 - 2y)(x^4 + 2x^2y + 4y^2)\).

Key Concepts

Difference of Cubes

The concept of a difference of cubes is quite fascinating in algebra because it allows us to factor expressions that may look complex at first glance. The difference of cubes refers to an expression in the form of \( a^3 - b^3 \). This specific pattern is captured by a special formula:

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

In this context, you look to express each term as a cube of another term. For instance, in our original problem the expression \( x^6 - 8y^3 \) is initially rewritten to resemble cubes: \( (x^2)^3 - (2y)^3 \). Using this method transforms the problem, making it easier to factor using the established formula. Once you identify and substitute \( a = x^2 \) and \( b = 2y \), the factors follow directly from the formula.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra, serving as a means to represent numbers, operations, and variables symbolically. Unlike simple arithmetic, algebra allows for working with unknowns and variables to explore mathematical relationships. Each expression contains terms which can be numbers, variables, or both, connected by various operations such as addition, subtraction, multiplication, and division.

• In our exercise, the expression \( x^6 - 8y^3 \) is composed of two terms.
• The terms are separated by a subtraction sign, making it a binomial.

Understanding the structure of algebraic expressions is crucial when performing operations such as expansion, simplification, and factoring. Factoring particularly requires recognizing patterns among terms to simplify or break down the expression into products of simpler expressions. This process is vital because it opens up new ways to solve equations and understand their properties.

Polynomial Factoring

Factoring polynomials is a fundamental technique in algebra that involves rewriting a polynomial as a product of simpler polynomials or expressions. This is akin to breaking down a number into its prime factors, but with variables and more complex algebraic structures.

• The key goal is to simplify expressions and equations, facilitating further solution or simplification steps.
• In the original problem, the polynomial expression \( x^6 - 8y^3 \) is factored using a specific algebraic identity for the difference of cubes.

The step-by-step solution involves first applying the difference of cubes formula, then simplifying the resulting expression. This typically involves:

• Expanding and rearranging the terms, as seen in the conversion to \((x^4 + 2x^2y + 4y^2)\).
• Checking if the resulting factors can be broken down further, ensuring the factorization is complete.

Being thorough in this process is crucial, as it ensures a correct and fully factored expression, which can then be used in more advanced mathematical analyses or in solving equations involving these polynomials.