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Chapter 6: Problem 3

Factor the given expressions completely. $$x^{3}+1$$

Short Answer

Expert verified
The factored form is \((x + 1)(x^2 - x + 1)\).

Step by step solution

01

Identify the Sum of Cubes

The expression \(x^3 + 1\) can be recognized as a sum of cubes. The sum of cubes formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(a = x\) and \(b = 1\), since \(1 = 1^3\).
02

Apply the Sum of Cubes Formula

Substitute \(a = x\) and \(b = 1\) into the sum of cubes formula. This gives: \((x + 1)((x)^2 - (x)(1) + (1)^2)\).
03

Simplify the Expression

Simplify the terms within the parentheses: \((x)^2 = x^2\), \((x)(1) = x\), and \((1)^2 = 1\). The expression now becomes \((x + 1)(x^2 - x + 1)\).
04

Verify the Factored Form

Multiply the factors back together to verify correctness: \((x + 1)(x^2 - x + 1) = x(x^2 - x + 1) + 1(x^2 - x + 1)\), which simplifies to \(x^3 - x^2 + x + x^2 - x + 1 = x^3 + 1\). This confirms that the factorization is correct.

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Key Concepts

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

Sum of Cubes
When you encounter an expression like \(x^3 + 1\), it's important to recognize it as a **Sum of Cubes**. This is a form where two perfect cubes are being added together. In our expression, \(x^3\) is the cube of \(x\), and \(1\) is the cube of the number \(1\), since \(1^3 = 1\). Understanding the structure of the sum of cubes can help you factor the expression efficiently. The formula to factor a sum of cubes is:
  • \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
To apply this to \(x^3 + 1\), set \(a = x\) and \(b = 1\). Recognizing and applying the sum of cubes formula is crucial for simplifying such polynomial expressions and is a useful algebraic technique.
Polynomials
Polynomials, like \(x^3 + 1\), are expressions that consist of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents of variables. Understanding the structure of polynomials is important, as they come in various forms:
  • A polynomial can be as simple as \(x + 1\) or as complex as \(x^3 - 5x^2 + 4x - 2\).
  • The terms of a polynomial are usually ordered from the highest degree to the lowest.
In our specific exercise, we focus on a cubic polynomial, as the highest power of \(x\) is 3. Mastery over polynomials allows for managing expressions in more complex algebraic problems and is a foundational element in algebra.
Factoring Techniques
**Factoring Techniques** involve rewriting an expression as a product of simpler expressions, or factors. For \(x^3 + 1\), the approach involves using the sum of cubes, one of the key factoring techniques in algebra. Factoring is beneficial because:
  • It simplifies expressions, making calculations easier.
  • It helps solve equations, where you set the expression to zero.
The step-by-step breakdown involves substituting values of \(a\) and \(b\) into the sum of cubes formula, which splits the cubic expression into the multiplied terms \((x + 1)(x^2 - x + 1)\). This not only simplifies solving but also verifies the correctness of the process. Mastering different factoring techniques enables smoother navigation through diverse mathematical challenges.

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Most popular questions from this chapter

Perform the indicated operations. Each expression occurs in the indicated area of application. $$\frac{\frac{L}{C}+\frac{R}{s C}}{s L+R+\frac{1}{s C}} \text { (electronics) }$$

Set up appropriate equations and solve the given stated problems. All numbers are accurate to at least two significant digits. A fox, pursued by a greyhound, has a start of 60 leaps. He makes 9 leaps while the greyhound makes but \(6 ;\) but, 3 leaps of the greyhound are equivalent to 7 of the fox. How many leaps must the greyhound make to overcome the fox? (Source: Elementary Algebra: Embracing the First Principles of the Science by Charles Davies, A.S. Barnes \& Company, 1852) (Hint: Let the unit of distance be one fox leap.)

Give the required explanations. Factor \(n^{3}-n,\) and then explain why it represents a multiple of 6 if \(n\) is an integer greater than 1

Reduce each fraction to simplest form. Each is from the indicated area of application. \(\frac{v^{2}-v_{0}^{2}}{v t-v_{0} t} \quad\) (rectilinear motion)

Solve for the indicated variable. Each equation comes from the technical area indicated. \(S q+S p=S p q+p,\) for \(q \quad\) (computer design)
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