Problem 65 - Algebra and Trigonometry

Question

Factor the expression completely. $$ x^{6}-8 y^{3} $$

Step-by-Step Solution

Answer

The expression factors to \((x^2 - 2y)(x^4 + 2x^2y + 4y^2)\).

Step 1: Recognize the Expression

The expression given is a difference of powers: \( x^6 - 8y^3 \). Notice that both terms are perfect powers, where \( x^6 \) is a power of \( x^2 \) and \( 8y^3 = (2y)^3 \). This indicates a possible structure for factoring.

Step 2: Use the Difference of Cubes Formula

The difference of cubes formula is \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \). We can transform our expression to fit this by rewriting \( x^6 \) as \( (x^2)^3 \) and \( 8y^3 \) as \( (2y)^3 \). Thus, we have \((x^2)^3 - (2y)^3\).

Step 3: Apply the Difference of Cubes Formula

Applying the difference of cubes formula, set \( a = x^2 \) and \( b = 2y \). Therefore, \((x^2)^3 - (2y)^3 = (x^2 - 2y)((x^2)^2 + x^2 \, (2y) + (2y)^2)\).

Step 4: Simplify the Factored Form

Simplify the expression inside the parenthesis: \((x^2)^2 + x^2 \, (2y) + (2y)^2\) becomes \(x^4 + 2x^2y + 4y^2 \). Thus, the completely factored form is \((x^2 - 2y)(x^4 + 2x^2y + 4y^2)\).

Key Concepts

Difference of Cubes

The difference of cubes is a special technique used to factor expressions that follow a specific form. When you have an expression like \(a^3 - b^3\), it can be factored into \((a - b)(a^2 + ab + b^2)\). This pattern helps simplify complex polynomials into more manageable pieces.

Why do we call it "difference of cubes"? A cube means raising a number or variable to the power of three, like in \(x^3\). A difference implies that we are subtracting one cube from another. This format only appears with subtraction: \((a^3 - b^3)\).

In our original exercise, we have \(x^6 - 8y^3\). Notice that \(x^6\) is the same as \((x^2)^3\); it's the cube of \(x^2\). Similarly, \(8y^3\) can be rewritten as \((2y)^3\). This matches the difference of cubes form, allowing us to apply the formula for easier factoring.

Understanding the difference of cubes formula provides a powerful tool in polynomial math. You use it whenever you see a cubic term subtracted from another, and it simplifies the polynomial down to more manageable terms.

Polynomial Expressions

Polynomial expressions are mathematical phrases involving sums and products of variables and coefficients raised to whole number powers. A polynomial can have constants, variables, and exponents that are non-negative integers.

In our exercise, the expression \(x^6 - 8y^3\) is a polynomial made up of two terms: \(x^6\) and \(-8y^3\). Each term is composed of a base raised to an exponent. The term \(x^6\) involves the variable \(x\) and the constant \(-8y^3\) involves both a variable, \(y\) and a numerical coefficient, 8.

It's essential to understand the structure of polynomial expressions to factor them effectively. The degree of a polynomial is determined by the highest power of the variable in the expression. In this case, the degree is 6 because the term \(x^6\) contains the highest power of \(x\).

Polynomials come in various forms, but recognizing the type of polynomial expression you're dealing with is crucial for choosing the right method to factor it. Whether dealing with the difference of cubes or any other polynomial, familiarity with expressions helps streamline solutions and improves problem-solving skills.

Factoring Techniques

Factoring techniques are strategies used to rewrite expressions as a product of simpler factors. These techniques make algebraic expressions easier to work with and solve.

Several techniques exist, such as:

• The Greatest Common Factor (GCF): Extracts the largest common factor from each term.
• Grouping: Involves grouping terms to simplify and factor.
• Difference of Squares: Applies when two squares are subtracted.
• Difference of Cubes: As discussed earlier, applies to expressions like \(a^3 - b^3\).

In the given exercise, we applied the difference of cubes factoring technique. The formula \((a^3 - b^3) = (a - b)(a^2 + ab + b^2)\) helped us factor \((x^2)^3 - (2y)^3\). Understanding which technique suits your problem is essential for efficient factoring.

Factoring is not just about memorizing formulas. It's about recognizing patterns and knowing which strategy to use in various scenarios. This skill is foundational in algebra and helps simplify expressions for solving equations or understanding their properties.