Problem 39 - Elementary and Intermediate Algebra

Question

Factor. $$ 64 x^{3}-27 y^{3} $$

Step-by-Step Solution

Answer

The factored form is \((4x-3y)(16x^2 + 12xy + 9y^2)\).

Step 1: Identify the Form of the Expression

The given expression is a binomial with two terms: \(64x^3\) and \(27y^3\). We notice that both terms are perfect cubes.

Step 2: Recognize the Formula for a Difference of Cubes

The formula for factoring a difference of cubes is \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Our goal is to rewrite \(64x^3 - 27y^3\) in this form.

Step 3: Identify \(a\) and \(b\) in the Difference of Cubes Formula

In the expression \(64x^3\), the cube root is \((4x)^3\), hence \(a = 4x\). In the term \(27y^3\), the cube root is \((3y)^3\), so \(b = 3y\).

Step 4: Substitute \(a\) and \(b\) into the Difference of Cubes Formula

Substitute \(a = 4x\) and \(b = 3y\) into the formula: \[(4x-3y)((4x)^2 + (4x)(3y) + (3y)^2)\].

Step 5: Simplify the Resulting Expression

Simplify each part of the expression:1. \((4x)^2 = 16x^2\)2. \((4x)(3y) = 12xy\)3. \((3y)^2 = 9y^2\).Thus, the expression becomes \((4x-3y)(16x^2 + 12xy + 9y^2)\).

Key Concepts

Understanding Binomials

In algebra, a **binomial** is a type of polynomial that consists of exactly two terms. These terms are connected by either a plus or a minus sign. Binomials are very common in algebraic expressions and are a critical concept to understand when learning polynomial arithmetic. Consider the example:

• The expression \(64x^3 - 27y^3\) is a binomial because it contains two distinct terms: \(64x^3\) and \(-27y^3\).
• The terms in a binomial can involve variables raised to a power, as well as coefficients (numbers multiplying the variables).

Mastering binomials is essential for higher-level mathematics, including operations like factoring, expansion, and solving equations involving binomials. Recognizing this simple structure helps in applying the appropriate factoring formulas efficiently.

Recognizing Perfect Cubes

A **perfect cube** is a number that can be expressed as the cube of an integer or a variable raised to the third power. Perfect cubes are essential in algebra, particularly when dealing with the difference of cubes or sum of cubes.For instance:

• The number 64 is a perfect cube because it equals \(4^3\).
• Similarly, the term \(x^3\) is a perfect cube since it is \((x)^3\).
• For the expression \(64x^3\), \(4x\) is the base that when cubed gives the term: \((4x)^3 = 64x^3\).

Knowing how to identify perfect cubes in an expression allows you to use specialized formulas to simplify the expressions further, like spotting a difference of cubes or sum of cubes formula that fits the situation. Recognizing these patterns helps in breaking down complex expressions into simpler, more manageable parts.

Working with Algebraic Expressions

**Algebraic expressions** are mathematical phrases that can include numbers, variables, and operation symbols. They form the foundation of algebra and are used for modeling real-world situations, solving equations, and more.Key points to understand:

• Algebraic expressions can range from simple monomials like \(3x\) to complex polynomials like \(64x^3 - 27y^3\).
• These expressions can be manipulated in various ways, such as by factoring, which involves expressing the equation as a product of its factors.
• When working with algebraic expressions, especially those involving perfect cubes and binomials, understanding and applying the appropriate formulas (like factoring the difference of cubes) becomes crucial.

Building a strong grasp of algebraic expressions paves the way for tackling more advanced mathematics topics, enabling one to convert verbal statements into mathematical equations and solve them efficiently.