Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 6: Problem 39
Factor. $$ 64 x^{3}-27 y^{3} $$
Short Answer
Expert verified
The factored form is \((4x-3y)(16x^2 + 12xy + 9y^2)\).
Step by step solution
01
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.
02
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.
03
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 \(b = 3y\).
04
Substitute \(a\) and \(b\) into the Difference of Cubes Formula
Substitute \(a = 4x\) and \(b = 3y\) into the formula: \[(4x-3y)((4x)^2 + (4x)(3y) + (3y)^2)\].
05
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 \((4x-3y)(16x^2 + 12xy + 9y^2)\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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).
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: \((4x)^3 = 64x^3\).
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.
