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Chapter 5: Problem 6

A student incorrectly factored. $$ x^{3}+8 \text { as }(x+2)^{3} $$

Short Answer

Expert verified
The correct factorization of \(x^3 + 8\) is \((x + 2)(x^2 - 2x + 4)\).

Step by step solution

01

Recognize the Given Expression

The expression provided is \(x^3 + 8\). It is important to recognize any patterns or forms it might resemble.
02

Identify the Correct Factoring Form

Observe that the given expression, \(x^3 + 8\), is a sum of cubes. The correct factoring formula for a sum of cubes is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
03

Apply the Sum of Cubes Formula

In this case, \(x^3 + 8\) can be rewritten as \(x^3 + 2^3\). By applying the sum of cubes formula where \(a = x\) and \(b = 2\), it becomes \((x + 2)(x^2 - 2x + 4)\).
04

Verify the Factored Form

Double-check the factored form by expanding \((x + 2)(x^2 - 2x + 4)\) to ensure it matches the original expression \(x^3 + 8\).

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Key Concepts

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

Sum of Cubes
A sum of cubes looks like this: \( a^3 + b^3 \). Each term is a perfect cube. In our exercise, the expression was \( x^3 + 8 \). Recognizing that 8 is \( 2^3 \) helps us identify this as a sum of cubes. To factor a sum of cubes, we use the formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here,\( a \) and \( b \) are the cube roots of the terms. For \( x^3 + 8 \), set \( a = x \) and \( b = 2 \). This gives the factored form: \( (x + 2)(x^2 - 2x + 4)\). By plugging in these values, we simplify the expression. One way to check our result is to expand it back to ensure it matches the original expression.
Factoring Techniques
Factoring is an important technique in algebra that helps simplify expressions and solve equations. Knowing different factoring methods allows you to recognize patterns and apply the correct formula. Here are some common techniques:
  • Factoring out the greatest common factor (GCF)
  • Factoring by grouping
  • Difference of squares
  • Quadratic trinomials
  • Sum and difference of cubes
In our exercise, we used the formula for the sum of cubes: \( a^3 + b^3 \). By identifying the form and applying the proper formula, we simplified the expression. Remember, checking your work by re-expanding the factored form saves you from mistakes and ensures accuracy.
Algebraic Expressions
Algebraic expressions combine numbers, variables, and operations. Factoring them makes complex calculations simpler. The key is understanding the structure of the expression. In the exercise, we dealt with \( x^3 + 8 \), recognized it as a sum of cubes, and then factored it. When working with algebraic expressions:
  • Identify potential patterns like common factors or special forms (difference of squares, sum of cubes, etc.)
  • Apply appropriate factoring techniques
  • Verify your results by expanding the factors
These steps will help you handle more complex problems efficiently. Practice different types of problems to become comfortable with recognizing various forms and applying the correct methods.

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