Factoring 8x³ - 125 A Step-by-Step Guide

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Factoring cubic expressions, like the one presented in the form 8x³ - 125, can seem daunting at first. However, by understanding and applying specific algebraic identities, these expressions can be simplified into more manageable forms. This article provides a detailed, step-by-step approach to factoring 8x³ - 125, highlighting the use of the difference of cubes formula and explaining each stage of the process to ensure clarity and comprehension.

Understanding the Difference of Cubes Formula

At the heart of factoring 8x³ - 125 lies the difference of cubes formula. This formula is a specific case of polynomial factorization that allows us to break down expressions in the form a³ - b³ into a product of simpler terms. The formula is expressed as:

a³ - b³ = (a - b)(a² + ab + b²)

This identity is essential for factoring expressions where two perfect cubes are subtracted. Recognizing this pattern is the first step in effectively factoring such expressions. The formula decomposes the difference of two cubes into a binomial (a - b) and a trinomial (a² + ab + b²), which are often easier to handle and analyze. Understanding this formula is crucial, not only for this specific problem but also for various algebraic manipulations and simplifications in mathematics.

To successfully apply the difference of cubes formula, it's imperative to identify the terms 'a' and 'b' correctly within the given expression. This involves recognizing perfect cubes, which are numbers or variables raised to the power of 3. In our case, 8x³ and 125 are both perfect cubes, which makes the difference of cubes formula perfectly applicable. Mastering this recognition and application process is a foundational skill in algebra, enabling one to tackle more complex problems with confidence.

Step-by-Step Factoring of 8x³ - 125

To factor 8x³ - 125 using the difference of cubes formula, we need to identify 'a' and 'b' such that a³ = 8x³ and b³ = 125. This identification is crucial as it sets the stage for applying the formula correctly. Let's break down the process:

  1. Identifying 'a' and 'b': The cube root of 8x³ is 2x because (2x)³ = 8x³. Thus, 'a' is 2x. The cube root of 125 is 5 because 5³ = 125. Thus, 'b' is 5. Correctly identifying 'a' and 'b' is a critical initial step as any error here will propagate through the rest of the solution.

  2. Applying the Formula: Now that we've identified 'a' as 2x and 'b' as 5, we can substitute these values into the difference of cubes formula:

    a³ - b³ = (a - b)(a² + ab + b²)

    Substituting gives us:

    8x³ - 125 = (2x - 5)((2x)² + (2x)(5) + 5²)

    This substitution is a direct application of the formula and transforms the original expression into a more factored form. The binomial term (2x - 5) represents the difference of the cube roots, while the trinomial term represents the sum of the squares and the product of the cube roots.

  3. Simplifying the Expression: Next, we simplify the expression by performing the operations within the parentheses:

    (2x - 5)(4x² + 10x + 25)

    Here, (2x)² simplifies to 4x², (2x)(5) simplifies to 10x, and 5² simplifies to 25. This simplification is crucial for arriving at the final factored form. Each term in the trinomial is carefully computed to ensure accuracy.

Thus, the factored form of 8x³ - 125 is (2x - 5)(4x² + 10x + 25). This final factored form is a product of a binomial and a trinomial, which cannot be factored further using real numbers. This completes the factoring process, providing a simplified representation of the original cubic expression.

Analyzing the Answer Choices

Now, let's analyze the answer choices provided in the question. The correct factorization of 8x³ - 125 is (2x - 5)(4x² + 10x + 25). We will examine each option to see which one matches our result.

  • A. (2x + 5)(4x² + 10x - 25): This option has incorrect signs. The binomial term should be (2x - 5), and the last term in the trinomial should be +25, not -25. Therefore, this choice is incorrect.

  • B. (2x + 5)(4x² - 10x + 25): This option also has incorrect signs. The binomial term should be (2x - 5). Therefore, this choice is incorrect.

  • C. (2x - 5)(4x² - 10x + 25): This option has an incorrect sign in the trinomial. The middle term should be +10x, not -10x. Therefore, this choice is incorrect.

  • D. (2x - 5)(4x² + 10x + 25): This option matches our factored form exactly. The binomial term (2x - 5) and the trinomial term (4x² + 10x + 25) are both correct. Therefore, this is the correct answer.

By systematically comparing each option with our derived factorization, we can confidently identify the correct answer. This process highlights the importance of careful algebraic manipulation and accurate sign management when factoring expressions. This analytical approach not only helps in selecting the right answer but also reinforces the understanding of the underlying algebraic principles.

Common Mistakes to Avoid

When factoring the difference of cubes, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help prevent errors and ensure accurate factoring. Here are some typical mistakes to watch out for:

  1. Incorrectly Applying the Formula: The difference of cubes formula, a³ - b³ = (a - b)(a² + ab + b²), has a specific structure. A common mistake is mixing up the signs or terms in the trinomial factor. For example, some might incorrectly write (a² - ab + b²) or (a² + ab - b²). Always double-check the formula and ensure the correct signs are used. The trinomial should always have the form (a² + ab + b²) when factoring the difference of cubes.

  2. Sign Errors: Sign errors are frequent, especially when dealing with the binomial factor (a - b) and the middle term ab in the trinomial. Ensure that the binomial factor correctly subtracts 'b' from 'a', and the middle term in the trinomial correctly adds the product of 'a' and 'b'. A simple sign mistake can completely change the factored form, leading to an incorrect answer. Careful attention to detail is essential to avoid these errors.

  3. Misidentifying 'a' and 'b': Incorrectly identifying 'a' and 'b' is another common mistake. This usually happens when the cube roots of the terms are not correctly determined. For example, in the expression 8x³ - 125, mistaking the cube root of 8x³ as 4x or the cube root of 125 as 25 will lead to incorrect values for 'a' and 'b'. Always take the time to correctly identify the cube roots before proceeding with the formula application.

  4. Forgetting to Simplify: After applying the formula, it's crucial to simplify the expression. This involves squaring terms, multiplying them, and combining like terms. Failing to simplify can result in an incomplete factorization. For instance, not simplifying (2x)² to 4x² or not calculating (2x)(5) as 10x can leave the expression in a non-final form. Always ensure that each term is fully simplified to reach the correct factored form.

By keeping these common mistakes in mind, you can improve your accuracy and confidence when factoring the difference of cubes. Careful attention to the formula, signs, term identification, and simplification will help you avoid these errors and arrive at the correct solution.

Conclusion

In conclusion, factoring 8x³ - 125 involves recognizing the expression as a difference of cubes and applying the corresponding formula. The step-by-step process includes identifying 'a' and 'b', substituting these values into the formula, and simplifying the resulting expression. The correct factorization is (2x - 5)(4x² + 10x + 25).

Avoiding common mistakes such as sign errors and misapplication of the formula is crucial for accurate factoring. By understanding and practicing these techniques, you can confidently factor cubic expressions and solve related algebraic problems. This skill is fundamental in algebra and has wide-ranging applications in mathematics and beyond. Mastering factoring techniques enhances your problem-solving abilities and provides a solid foundation for more advanced mathematical concepts.