Factoring By Grouping Determining Factors Of 12x³ - 2x² + 18x - 3

Factoring polynomials can sometimes seem like navigating a complex maze, but with the right strategies, even seemingly intricate expressions can be broken down into simpler, manageable components. One powerful technique in our factoring toolkit is factoring by grouping, a method particularly effective when dealing with polynomials containing four or more terms. In this comprehensive guide, we will delve deep into the concept of factoring by grouping, using the example of the polynomial 12x³ - 2x² + 18x - 3 to illustrate the process step-by-step. We'll explore how to identify opportunities for grouping, extract common factors, and ultimately express the polynomial as a product of simpler expressions. By understanding and mastering factoring by grouping, you'll significantly enhance your ability to manipulate and solve algebraic equations. Let's embark on this journey to unlock the secrets of factoring by grouping!

Understanding Factoring by Grouping

Before we dive into the specifics of factoring 12x³ - 2x² + 18x - 3, let's lay a solid foundation by understanding the core principles of factoring by grouping. This technique is most effective when dealing with polynomials that have an even number of terms, typically four or more. The fundamental idea is to strategically pair terms together, identify common factors within each pair, and then factor out these common factors. If done correctly, this process will reveal a common binomial factor across the resulting expressions, which can then be factored out, leading to the complete factorization of the original polynomial.

Why does factoring by grouping work? It's based on the distributive property in reverse. The distributive property states that a(b + c) = ab + ac. Factoring is essentially the process of undoing this distribution. By grouping terms and factoring out common factors, we're aiming to reverse this process and express the polynomial as a product of factors. Factoring by grouping allows us to systematically identify and extract these common factors, even when they are not immediately obvious in the original polynomial expression. This systematic approach makes factoring by grouping a valuable tool in our algebraic arsenal.

To illustrate this, consider a general example: ax + ay + bx + by. We can group the first two terms and the last two terms: (ax + ay) + (bx + by). Now, factor out the common factor from each group: a(x + y) + b(x + y). Notice that (x + y) is a common binomial factor. Factoring this out, we get: (x + y)(a + b). This demonstrates the basic mechanism of factoring by grouping. Now, let's apply this to our specific example.

Applying Factoring by Grouping to 12x³ - 2x² + 18x - 3

Now, let's apply the principles of factoring by grouping to our polynomial: 12x³ - 2x² + 18x - 3. The first step is to group the terms. A natural grouping here is to pair the first two terms and the last two terms: (12x³ - 2x²) + (18x - 3). This grouping strategy is often the most intuitive starting point, but it's important to remember that other groupings might be possible, and sometimes necessary, to achieve successful factorization. The key is to look for pairs of terms that share a common factor.

Next, we identify and factor out the greatest common factor (GCF) from each group. In the first group, (12x³ - 2x²), the GCF is 2x². Factoring this out, we get: 2x²(6x - 1). In the second group, (18x - 3), the GCF is 3. Factoring this out, we get: 3(6x - 1). So, our expression now looks like this: 2x²(6x - 1) + 3(6x - 1). This is a crucial step in the process, as it sets the stage for the final factorization.

Notice that we now have a common binomial factor: (6x - 1). This is the key to completing the factorization. We can factor out this common binomial factor from the entire expression: (6x - 1)(2x² + 3). And that's it! We have successfully factored the polynomial 12x³ - 2x² + 18x - 3 by grouping. The factored form is (6x - 1)(2x² + 3). This resulting expression represents the product of two polynomials, demonstrating the effectiveness of factoring by grouping.

Analyzing the Options and Identifying the Correct Grouping

Now, let's examine the options provided in the original question and see which one demonstrates a correct step in the process of factoring by grouping 12x³ - 2x² + 18x - 3.

Option A: 2x²(6x - 1) + 3(6x - 1)

This option perfectly matches the intermediate step we arrived at when factoring out the GCF from each group. As we demonstrated earlier, factoring 2x² from (12x³ - 2x²) gives us 2x²(6x - 1), and factoring 3 from (18x - 3) gives us 3(6x - 1). This clearly shows the correct application of factoring out the GCF from each group, leading to the identification of the common binomial factor (6x - 1). Therefore, Option A is a correct representation of a step in factoring by grouping.

Option B: 2x²(6x - 1) - 3(6x - 1)

This option is similar to Option A, but it has a crucial difference: the subtraction sign. While the first term, 2x²(6x - 1), is correct, the second term, -3(6x - 1), is incorrect. If we were to distribute the -3, we would get -18x + 3, which is the negative of our original terms (18x - 3). This indicates an error in factoring out the GCF or in maintaining the correct signs. Therefore, Option B is incorrect.

Option C: 6x(2x² - 3) - 1(2x² - 3)

This option represents a different grouping strategy. Let's analyze if this grouping is valid. If we distribute, we get 12x³ - 18x - 2x² + 3. Rearranging the terms, we get 12x³ - 2x² - 18x + 3. Notice that the sign of the 18x term is incorrect compared to our original polynomial. This indicates that this grouping strategy, while perhaps tempting, does not lead to a correct factorization. Therefore, Option C is incorrect.

Option D: 6x(2x² + 3) + 1(2x² + 3)

Similar to Option C, this option presents an alternative grouping. Distributing the terms, we get 12x³ + 18x + 2x² + 3. Rearranging the terms, we have 12x³ + 2x² + 18x + 3. Comparing this to our original polynomial 12x³ - 2x² + 18x - 3, we see that the sign of the 2x² term and the constant term are incorrect. This confirms that this grouping does not lead to a correct factorization of the original polynomial. Therefore, Option D is incorrect.

Conclusion: Mastering Factoring by Grouping

In conclusion, Option A, 2x²(6x - 1) + 3(6x - 1), accurately shows one way to determine the factors of 12x³ - 2x² + 18x - 3 by grouping. This option demonstrates the crucial step of factoring out the greatest common factor from each group, leading to the identification of the common binomial factor. By mastering the technique of factoring by grouping, you gain a valuable tool for simplifying and solving algebraic expressions. Remember to group terms strategically, identify and factor out the GCF from each group, and look for the emergence of a common binomial factor. With practice, you'll become adept at recognizing opportunities for factoring by grouping and efficiently breaking down complex polynomials into their simpler components. Factoring by grouping isn't just a mathematical technique; it's a skill that enhances your problem-solving abilities and deepens your understanding of algebraic structures. Keep practicing, and you'll unlock the power of factoring by grouping!

This guide has provided a comprehensive explanation of factoring by grouping, using the example of 12x³ - 2x² + 18x - 3. By understanding the underlying principles and applying the steps systematically, you can confidently tackle a wide range of factoring problems. Remember, the key is to practice and develop your intuition for identifying the most effective grouping strategies. Happy factoring!