Problem 45 - Algebra for College Students

Question

Use the sum-of-two-cubes or the difference-of-two-cubes pattern to factor each of the following. $$a^{3}-64$$

Step-by-Step Solution

Answer

The expression factors as \( (a-4)(a^2+4a+16) \).

Step 1: Identify the Problem Type

The expression given is a cubic expression: \( a^3 - 64 \). Notice that 64 is a perfect cube (since \( 64 = 4^3 \)), and recognize this as a difference of two cubes.

Step 2: Recall the Formula for Factoring Difference of Cubes

The difference of cubes formula is: \( x^3 - y^3 = (x-y)(x^2 + xy + y^2) \). Recognize this formula will be used where \( x = a \) and \( y = 4 \).

Step 3: Apply the Difference of Two Cubes Formula

Substitute \( x = a \) and \( y = 4 \) into the difference of cubes formula. This yields: \( a^3 - 4^3 = (a - 4)(a^2 + 4a + 16) \).

Step 4: Verification

Verify by expanding \( (a - 4)(a^2 + 4a + 16) \) to ensure it returns the original cubic expression, \( a^3 - 64 \). The expansion gives \( a^3 + 4a^2 + 16a - 4a^2 - 16a - 64 \) which simplifies to \( a^3 - 64 \). This confirms our factorization is correct.

Key Concepts

Difference of Cubes

The concept of the difference of cubes is a powerful technique used in factoring polynomials, particularly when you encounter expressions like \( a^3 - b^3 \). This type of polynomial can be rewritten in a factored form using a specific formula. Recognizing this pattern involves identifying each term as a perfect cube. For example, in the exercise \( a^3 - 64 \), notice that 64 can be expressed as \( 4^3 \), making it evident that the expression is a difference of cubes.

The formula for factoring the difference of cubes is given by:

• \( x^3 - y^3 = (x-y)(x^2 + xy + y^2) \)

In this formula, \( x \) and \( y \) are the cube roots of the terms \( a^3 \) and \( b^3 \). Applying this formula involves identifying \( x \) and \( y \) as the cube roots of the given terms, and directly substituting them into the formula. Understanding this approach is crucial for correct and efficient polynomial factorization.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. These play a fundamental role in algebra and are foundational in solving equations, factoring, and simplifying expressions. In the context of factoring polynomials, algebraic expressions like \( a^3 - 64 \) are shown as a way to simplify and manipulate expressions into product forms.

When working with algebraic expressions, it's crucial to:

• Recognize patterns (e.g., perfect square trinomials, sum and difference of cubes)
• Understand the operations involved (addition, subtraction, multiplication)
• Use substitution and simplification techniques effectively

By mastering these concepts, you can approach problems involving polynomial expressions with confidence and ease. This, in turn, aids in broader problem-solving and computation skills.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials. This technique is crucial in simplifying complex expressions and solving polynomial equations. Factorization can take many forms depending on the polynomial in question, such as recognizing patterns like the difference of squares or cubes.

To factor polynomials, follow these steps:

• Identify the type of polynomial pattern (e.g., difference of cubes)
• Apply the appropriate factoring formula or technique
• Verify the factorization by expanding the factors to check against the original polynomial

In the exercise \( a^3 - 64 \), the process involved recognizing \( a^3 - 64 \) as a difference of cubes and applying its specific factor formula. This results in a factored form \( (a-4)(a^2+4a+16) \). Always remember to check your work by expanding it back to ensure accuracy. This skill not only aids in algebra but also lays the groundwork for further calculus and higher mathematics applications.