Factoring By Grouping Determining Factors Of X³ + 5x² - 6x - 30

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Factoring polynomials is a fundamental skill in algebra, and one powerful technique for accomplishing this is factoring by grouping. This method is particularly useful when dealing with polynomials with four or more terms. In this article, we will delve into the process of factoring by grouping, using the example polynomial x³ + 5x² - 6x - 30. We will explore the underlying principles, step-by-step procedures, and common pitfalls to avoid. By the end of this guide, you'll be well-equipped to tackle similar factoring problems with confidence.

Understanding Factoring by Grouping

Factoring by grouping is a technique used to factor polynomials with four or more terms. The basic idea is to group terms together in such a way that you can factor out a common factor from each group. This process often reveals a common binomial factor that can then be factored out, leading to the complete factorization of the polynomial. The key to successful factoring by grouping lies in identifying the right groupings and common factors.

In the context of the given polynomial, x³ + 5x² - 6x - 30, we aim to rewrite the expression in a form that allows us to extract common factors from pairs of terms. This is where the different options provided (A, B, C, and D) come into play. Each option represents a potential grouping and factoring step. Our goal is to determine which option correctly demonstrates the initial step in factoring the polynomial by grouping.

Core Concepts of Factoring by Grouping

Before we dive into the specific example, let's solidify the core concepts behind factoring by grouping:

  1. Grouping: The first step is to group the terms of the polynomial into pairs. This is typically done by looking for terms that share common factors.
  2. Factoring out Common Factors: From each group, identify and factor out the greatest common factor (GCF). This will leave you with a factored expression within each group.
  3. Identifying the Common Binomial Factor: After factoring out the GCF from each group, you should ideally be left with the same binomial factor in each group. This is the crucial step in factoring by grouping.
  4. Factoring out the Binomial Factor: Once you've identified the common binomial factor, factor it out from the entire expression. This will leave you with the fully factored polynomial.

Understanding these core concepts is essential for successfully applying the factoring by grouping technique. Now, let's apply these concepts to our example polynomial.

Analyzing the Options: A Step-by-Step Approach

To determine which option correctly shows the initial step in factoring x³ + 5x² - 6x - 30 by grouping, we will analyze each option individually. We will look for the correct grouping and factoring of the greatest common factor (GCF) from each group.

Option A: x(x²-5)+6(x²-5)

This option suggests grouping the terms as follows: (x³ - 6x) + (5x² - 30), and then factoring out x from the first group and 6 from the second group. Let's examine the original polynomial x³ + 5x² - 6x - 30. If we try to factor x from the first group, x³ - 6x, we get x(x² - 6). However, the option provides x(x² - 5), which is incorrect. Similarly, if we factor 6 from (-6x - 30), we should get -6(x + 5) or 6(-x - 5), not 6(x² - 5) as suggested in the option. Thus, Option A is incorrect because it misidentifies the common factors and does not accurately represent the factoring process for the given polynomial.

Option B: x(x²+5)-6(x²+5)

This option proposes a grouping that could lead to a common binomial factor. Let's see if this grouping arises from the original polynomial. Grouping the terms as (x³ + 5x²) + (-6x - 30), we can factor out x² from the first group, resulting in x²(x + 5). From the second group, we can factor out -6, resulting in -6(x + 5). Combining these, we get x²(x + 5) - 6(x + 5). Comparing this with Option B, we see a discrepancy. Option B has x(x² + 5) instead of x²(x + 5) for the first group. Therefore, Option B is incorrect as it does not accurately reflect the initial factoring step by grouping.

Option C: x²(x-5)+6(x-5)

This option suggests a common binomial factor of (x - 5). However, this doesn't align with the correct grouping and factoring of the original polynomial. As we analyzed in Option B, grouping the terms (x³ + 5x²) + (-6x - 30) leads to x²(x + 5) - 6(x + 5), which has a common binomial factor of (x + 5), not (x - 5). Therefore, Option C is incorrect because it presents an incorrect binomial factor, which does not arise from the proper grouping and factoring of the original expression.

Option D: x²(x+5)-6(x+5)

This option aligns perfectly with the correct factoring process. As we discussed in the analysis of Option B, grouping the terms of x³ + 5x² - 6x - 30 as (x³ + 5x²) + (-6x - 30) allows us to factor out x² from the first group and -6 from the second group. This gives us x²(x + 5) - 6(x + 5), which exactly matches Option D. This demonstrates the correct initial step in factoring by grouping. Thus, Option D is correct. The common binomial factor (x + 5) is now evident, and we can proceed with the final factoring step.

The Complete Factoring Process

Having identified the correct initial grouping and factoring step (Option D), let's complete the factoring process for x³ + 5x² - 6x - 30.

  1. Initial Grouping and Factoring: We have established that x³ + 5x² - 6x - 30 can be rewritten as x²(x + 5) - 6(x + 5).
  2. Factoring out the Common Binomial Factor: Now, we can see that (x + 5) is a common factor in both terms. Factoring out (x + 5), we get (x + 5)(x² - 6).

Therefore, the completely factored form of x³ + 5x² - 6x - 30 is (x + 5)(x² - 6).

Step-by-step breakdown

To further illustrate the complete factoring process, here's a step-by-step breakdown:

  1. Original Polynomial: x³ + 5x² - 6x - 30
  2. Group the terms: (x³ + 5x²) + (-6x - 30)
  3. Factor out the GCF from each group: x²(x + 5) - 6(x + 5)
  4. Factor out the common binomial factor: (x + 5)(x² - 6)
  5. Final Factored Form: (x + 5)(x² - 6)

This step-by-step process provides a clear roadmap for factoring by grouping, from the initial grouping to the final factored form.

Common Mistakes to Avoid

Factoring by grouping can be a straightforward technique, but there are common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate factoring.

  1. Incorrect Grouping: Choosing the wrong grouping can lead to dead ends. The key is to group terms that share common factors. If the initial grouping doesn't lead to a common binomial factor, try a different grouping.
  2. Incorrectly Factoring out the GCF: Ensure you factor out the greatest common factor from each group. Factoring out a smaller factor may leave you with a more complex expression to deal with.
  3. Missing the Sign: Pay close attention to the signs when factoring out the GCF, especially when factoring out a negative number. For example, factoring out -6 from (-6x - 30) requires careful attention to the signs to correctly get -6(x + 5).
  4. Not Factoring Completely: After factoring out the common binomial factor, check if the resulting quadratic factor can be further factored. In our example, (x² - 6) cannot be factored further using integer coefficients, but in other cases, you may need to apply additional factoring techniques.

By being mindful of these common mistakes, you can improve your accuracy and efficiency in factoring by grouping.

Practice Problems

To solidify your understanding of factoring by grouping, let's work through a few practice problems. These problems will give you the opportunity to apply the concepts and techniques we've discussed.

  1. Factor: 2x³ + 3x² + 4x + 6
  2. Factor: x³ - 4x² - 9x + 36
  3. Factor: 3x³ + 6x² - 5x - 10

Solutions:

  1. 2x³ + 3x² + 4x + 6 = (x² + 2)(2x + 3)
  2. x³ - 4x² - 9x + 36 = (x² - 9)(x - 4) = (x + 3)(x - 3)(x - 4)
  3. 3x³ + 6x² - 5x - 10 = (x² - 5)(3x + 6) = 3(x² - 5)(x + 2)

Working through these practice problems will help you build confidence and mastery in factoring by grouping.

Conclusion

In summary, factoring by grouping is a valuable algebraic technique for polynomials with four or more terms. By grouping terms, factoring out common factors, and identifying the common binomial factor, we can effectively factor complex polynomials. In the case of x³ + 5x² - 6x - 30, the correct initial grouping and factoring is demonstrated by Option D: x²(x + 5) - 6(x + 5), which leads to the complete factorization of (x + 5)(x² - 6). By understanding the underlying concepts, avoiding common mistakes, and practicing consistently, you can master this technique and enhance your algebraic skills.

This comprehensive guide has provided you with the knowledge and tools to tackle factoring by grouping problems. Remember, practice is key to mastering any mathematical concept, so continue to work through examples and challenge yourself with increasingly complex problems. With dedication and perseverance, you'll become proficient in factoring by grouping and excel in your algebraic endeavors. Factoring polynomials is a critical skill in algebra, serving as a foundation for more advanced topics. As you become more comfortable with factoring techniques like grouping, you'll unlock the ability to simplify expressions, solve equations, and tackle a wider range of mathematical challenges. Embrace the process, learn from your mistakes, and celebrate your successes along the way. The journey to mastering algebra is a rewarding one, and factoring by grouping is a significant milestone in that journey.