Factoring By Grouping X^3 - 12x^2 - 2x + 24 A Comprehensive Guide

Factoring polynomials is a crucial skill in algebra, and one effective technique for factoring higher-degree polynomials is factoring by grouping. This method involves strategically grouping terms within the polynomial and identifying common factors to simplify the expression. This article delves into the process of factoring the cubic polynomial $x^3 - 12x^2 - 2x + 24$ by grouping, providing a step-by-step explanation and highlighting the correct approach.

Understanding Factoring by Grouping

Factoring by grouping is a method used to factor polynomials with four or more terms. The basic idea is to group terms in pairs, factor out the greatest common factor (GCF) from each pair, and then look for a common binomial factor. This technique allows us to break down complex polynomials into simpler, manageable factors. The key to successful factoring by grouping lies in identifying the correct groupings and factoring out the GCF accurately. This process often involves trial and error, but with practice, it becomes more intuitive.

When approaching a polynomial for factoring by grouping, the initial step involves examining the coefficients and constants to determine potential groupings. We aim to group terms that share common factors, which can be either variables or constants. For instance, in the polynomial $x^3 - 12x^2 - 2x + 24$, we observe that the first two terms ($x^3$ and $-12x^2$) share a common factor of $x^2$, while the last two terms ($-2x$ and $+24$) share a common factor of $-2$. This observation guides us in forming the initial groupings. Furthermore, it's crucial to pay attention to the signs of the terms. Factoring out a negative sign can sometimes reveal a common binomial factor that might not be apparent otherwise. In this specific case, factoring out $-2$ from the last two terms is essential to obtain a common binomial factor with the first group. Factoring by grouping is not always straightforward, and some polynomials may require rearranging terms or trying different groupings before a common factor emerges. However, with a systematic approach and careful observation, this technique can be a powerful tool for simplifying and solving polynomial equations.

Step-by-Step Factoring of $x^3 - 12x^2 - 2x + 24$

Let's apply the factoring by grouping method to the given polynomial, $x^3 - 12x^2 - 2x + 24$. Here's a detailed breakdown of the steps involved:

1. Initial Grouping: The first step is to group the terms in pairs. A natural grouping based on the structure of the polynomial is to group the first two terms and the last two terms together:

   $(x^3 - 12x^2) + (-2x + 24)$

   This grouping strategy is based on the observation that the first two terms contain powers of $x$, while the last two terms involve a linear term and a constant. This arrangement sets the stage for identifying common factors within each group, a critical step in the factoring process. It's essential to consider the signs of the terms when grouping, ensuring that the subsequent factoring steps maintain the integrity of the original polynomial. In this case, the positive sign between the two groups indicates that we are adding the factored expressions, which aligns with the polynomial's overall structure. Correct grouping is not merely a mechanical step but a thoughtful decision that lays the foundation for the rest of the factoring process. An incorrect grouping can lead to a dead end, making it impossible to find a common binomial factor. Therefore, careful consideration of the terms' relationships and signs is crucial for successful factoring by grouping.

2. Factor out the Greatest Common Factor (GCF) from each group: Next, we identify and factor out the GCF from each group:

   From the first group $(x^3 - 12x^2)$, the GCF is $x^2$. Factoring this out, we get:

   $x^2(x - 12)$

   From the second group $(-2x + 24)$, the GCF is $-2$. Factoring this out, we get:

   $-2(x - 12)$

   Factoring out the GCF from each group is a pivotal step in the factoring by grouping method. It simplifies each group and reveals a potential common binomial factor, which is essential for the next stage of factoring. In the first group, $x^3 - 12x^2$, the GCF is $x^2$ because $x^2$ is the highest power of $x$ that divides both terms evenly. Factoring $x^2$ out leaves us with $x^2(x - 12)$. This step demonstrates the distributive property in reverse, where we are essentially undoing the multiplication of $x^2$ by $(x - 12)$. Similarly, in the second group, $-2x + 24$, the GCF is $-2$. Factoring out $-2$ is crucial because it transforms the expression inside the parentheses to $(x - 12)$, which is the same binomial factor we obtained from the first group. This alignment of binomial factors is the key to successful factoring by grouping. The decision to factor out a negative number from the second group is strategic, as it ensures that the binomial factors match. If we had factored out $+2$ instead of $-2$, we would have obtained $2(-x + 12)$, which does not directly match the $(x - 12)$ from the first group. Therefore, careful attention to the signs and the identification of the correct GCF are critical for progressing towards the final factored form.

3. Identify the Common Binomial Factor: Now, we rewrite the expression with the factored groups:

   $x^2(x - 12) - 2(x - 12)$

   Notice that $(x - 12)$ is a common binomial factor in both terms. This common binomial factor is the linchpin of the factoring by grouping method. Its presence indicates that the initial grouping and factoring of GCFs were successful. The ability to identify this common factor is a testament to the method's effectiveness in simplifying polynomials. In this context, the binomial $(x - 12)$ acts as a single entity that can be factored out, similar to how we factor out a common monomial factor. This step is not merely about recognizing the common binomial; it's about understanding its role in bridging the two factored groups into a single, cohesive factored expression. The common binomial factor serves as a unifying element, allowing us to consolidate the polynomial into a more compact form. Without this commonality, the factoring process would stall, highlighting the importance of strategic grouping and GCF extraction in the earlier steps.

4. Factor out the Common Binomial Factor: We factor out the common binomial factor $(x - 12)$:

   $(x - 12)(x^2 - 2)$

   Factoring out the common binomial factor is the culmination of the factoring by grouping process. It's the step where we consolidate the two separate groups into a single, factored expression. By treating $(x - 12)$ as a single entity, we apply the distributive property in reverse, effectively factoring it out from both terms. This process transforms the expression $x^2(x - 12) - 2(x - 12)$ into $(x - 12)(x^2 - 2)$. The remaining factor, $(x^2 - 2)$, is formed by the terms that were multiplied by the common binomial factor, in this case, $x^2$ and $-2$. This step not only simplifies the polynomial but also reveals its underlying structure, showing how the original terms are related through multiplication. The resulting expression, $(x - 12)(x^2 - 2)$, is the factored form of the original polynomial, $x^3 - 12x^2 - 2x + 24$, achieved through the strategic application of factoring by grouping. At this point, it's worth checking whether the resulting factors can be factored further. In this instance, $(x^2 - 2)$ cannot be factored further using elementary methods over the integers, so we have reached the final factored form.

5. Final Factored Form: The factored form of the polynomial $x^3 - 12x^2 - 2x + 24$ is:

   $(x - 12)(x^2 - 2)$

   The final factored form, $(x - 12)(x^2 - 2)$, represents the complete factorization of the original cubic polynomial $x^3 - 12x^2 - 2x + 24$ using the method of factoring by grouping. This result provides a concise and simplified representation of the polynomial, which can be valuable for various algebraic manipulations, such as finding roots, solving equations, and analyzing the polynomial's behavior. The factored form reveals that the polynomial has a linear factor $(x - 12)$ and a quadratic factor $(x^2 - 2)$. The linear factor corresponds to a real root at $x = 12$, while the quadratic factor can be further analyzed to find additional roots, which in this case would be irrational roots. The significance of achieving the final factored form lies not only in simplifying the expression but also in gaining insights into the polynomial's characteristics and solutions. The process of factoring transforms a complex expression into a product of simpler expressions, making it easier to understand and work with. In this instance, the factored form allows us to readily identify one of the roots of the polynomial and provides a foundation for further analysis if needed. Furthermore, the factored form can be used to verify the original expression by expanding the product of the factors, ensuring that we arrive back at the initial polynomial.

Identifying the Correct Option

Based on the steps above, the correct option that shows one way to determine the factors of $x^3 - 12x^2 - 2x + 24$ by grouping is:

D. $x^2(x - 12) - 2(x - 12)$

This option accurately represents the intermediate step where the GCFs have been factored out from each group, revealing the common binomial factor $(x - 12)$. This step is crucial in the factoring by grouping process, as it sets the stage for the final factorization. The other options either have incorrect groupings or incorrect factoring of the GCF, leading to a dead end in the factoring process. Option A, $x(x^2 - 12) + 2(x^2 - 12)$, groups the terms differently and does not align with the correct factoring steps. Option B, $x(x^2 - 12) - 2(x^2 - 12)$, also uses an incorrect grouping and does not lead to the proper common binomial factor. Option C, $x^2(x - 12) + 2(x - 12)$, while having the correct binomial factor, has an incorrect sign in the second group, which would lead to a different final factored form. Therefore, option D is the only option that accurately reflects the intermediate step in the factoring by grouping method for the given polynomial. It demonstrates the correct application of factoring out GCFs and sets the stage for the final step of factoring out the common binomial factor.

Conclusion

Factoring by grouping is a powerful technique for simplifying polynomials. By carefully grouping terms and identifying common factors, we can break down complex expressions into manageable factors. In the case of $x^3 - 12x^2 - 2x + 24$, the correct grouping and factoring lead to the expression $(x - 12)(x^2 - 2)$. Mastering this technique is essential for success in algebra and higher-level mathematics.

The ability to factor polynomials by grouping is a fundamental skill in algebra, providing a means to simplify expressions, solve equations, and gain deeper insights into polynomial functions. The process involves a series of strategic steps, including grouping terms, factoring out greatest common factors, identifying common binomial factors, and ultimately expressing the polynomial as a product of simpler factors. In the context of the cubic polynomial $x^3 - 12x^2 - 2x + 24$, we have demonstrated how this method can be applied systematically to arrive at the factored form $(x - 12)(x^2 - 2)$. This factored form not only simplifies the polynomial but also reveals its roots and provides a foundation for further analysis. The technique of factoring by grouping is not limited to cubic polynomials; it can be applied to polynomials of higher degrees and with varying numbers of terms, provided that a suitable grouping strategy can be identified. The key to success lies in careful observation, attention to detail, and a systematic approach. Mastering factoring by grouping enhances algebraic fluency and problem-solving skills, making it an indispensable tool for students and practitioners of mathematics.