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Chapter 6: Problem 14

Factor completely. $$ x^{3}-8 $$

Short Answer

Expert verified
(x - 2)(x^2 + 2x + 4)

Step by step solution

01

- Identify a Difference of Cubes

The given expression is in the form of a difference of cubes, where it can be written as \[ x^3 - 2^3 \]
02

- Apply the Difference of Cubes Formula

The formula for factoring the difference of cubes is given by \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] Here, \( a = x \) and \( b = 2 \).
03

- Substitute and Simplify

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring
Factoring is the process of breaking down a complex expression into simpler components, called factors. When we factor an expression, we are essentially reversing the multiplication process.

For example, in the given exercise, we are asked to factor the expression \( x^3 - 8 \). This expression is a difference of cubes, which can be factored using a specific formula:

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

By identifying our values of \( a \) and \( b \), we can break down the polynomial into simpler, easily understandable products.
Algebraic Expressions
An algebraic expression is a combination of variables, constants, and operations like addition, subtraction, multiplication, and division. In the expression \( x^3 - 8 \), we have a polynomial comprising the variables and constants.

Algebraic expressions can take many forms, and one of the most common tasks in algebra is to manipulate these expressions to simplify or solve them. Understanding how to break down complex expressions into simpler parts, like factoring polynomials, is a foundational skill.

For instance, recognizing that \( x^3 - 8 \) can be expressed as \( x^3 - 2^3 \) helps leverage the difference of cubes formula for simplification.
Polynomial Factorization
Polynomial factorization is a technique used to express a polynomial as a product of its factors. It helps in solving polynomial equations and simplifying expressions. The difference of cubes is a specific type of polynomial that allows for straightforward factorization.

Our example, \( x^3 - 8 \), fits into the structure of a difference of cubes. The steps to factor such polynomials include:
  • Identifying the cubes in the expression (\( x^3 \) and \( 2^3 \)).
  • Applying the appropriate formula, \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \], where \( a = x \) and \( b = 2 \).
  • Substituting back into the formula to obtain the factors: \[ x^3 - 2^3 = (x - 2)(x^2 + 2x + 4) \]
By mastering these steps, polynomial factorization becomes a powerful tool for solving a wide range of algebraic problems.

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Most popular questions from this chapter

Find the zeros of each function. $$ f(x)=x^{3}-2 x^{2}-x+2 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ x^{2}+2 x y+y^{2}-9 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 49 a^{4}+100 $$

What is wrong with solving \(x^{2}=3 x\) by dividing both sides of the equation by \(x ?\)

Akio concludes that since \(x^{2}-9=(x-3)(x+3)\) it must follow that \(x^{2}+9=(x+3)(x-3) .\) What mistake(s) is he making?
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