Factoring Expressions A Comprehensive Guide To 6x^2 - 8x + 21x - 28

Factoring expressions is a fundamental skill in algebra, serving as a cornerstone for solving equations, simplifying expressions, and tackling more advanced mathematical concepts. In this detailed guide, we will explore the process of factoring the expression $6x^2 - 8x + 21x - 28$ using various methods. Whether you're a student looking to master factoring or simply seeking a refresher, this comprehensive explanation will provide you with the knowledge and confidence to tackle similar problems.

Understanding Factoring

Before diving into the specifics of our expression, it's crucial to understand the basic principle of factoring. Factoring is the reverse process of multiplication. It involves breaking down an expression into its constituent factors, which, when multiplied together, yield the original expression. Think of it as finding the building blocks that make up a larger structure. For example, factoring the number 12 gives us 2 × 2 × 3, where 2 and 3 are the prime factors of 12.

In algebra, we factor expressions containing variables and constants. A common method for factoring expressions with four terms, like the one we're addressing, is factoring by grouping. This technique involves grouping terms together, identifying common factors within each group, and then extracting those factors to simplify the expression. Factoring by grouping is a powerful technique, particularly effective when dealing with expressions where a direct common factor across all terms isn't immediately apparent. This method leverages the distributive property in reverse, allowing us to break down complex expressions into simpler, more manageable parts. The core idea is to strategically pair terms in such a way that each group shares a common factor, paving the way for subsequent factorization. The success of this method often hinges on the judicious selection of pairs and a keen eye for identifying common factors. By systematically applying this approach, we can unravel intricate expressions and express them in their factored form, which is essential for solving equations and simplifying mathematical problems.

1. Initial Expression

Our starting point is the expression:

$$6x^2 - 8x + 21x - 28$$

This expression consists of four terms, which makes it suitable for factoring by grouping. Factoring by grouping is a technique that allows us to break down complex expressions into simpler factors. This method is particularly useful when dealing with polynomials that don't have a readily apparent common factor across all terms. The basic idea behind factoring by grouping is to rearrange and group terms in such a way that each group shares a common factor. By identifying and extracting these common factors, we can simplify the expression and ultimately factor it into a product of simpler expressions. This technique is a fundamental tool in algebra and is essential for solving equations, simplifying expressions, and understanding more advanced mathematical concepts.

2. Grouping Terms

The first step in factoring by grouping is to group the terms in pairs. We can group the first two terms and the last two terms together:

$$(6x^2 - 8x) + (21x - 28)$$

The grouping of terms is a crucial step in the factoring by grouping method. It involves strategically pairing terms within the expression to facilitate the identification of common factors. The way we group terms can significantly impact the ease and success of the factorization process. In some cases, different groupings may lead to the same factored form, but one grouping might be more straightforward than another. The key is to look for pairs of terms that share a common factor, whether it's a numerical factor, a variable, or a combination of both. By carefully selecting the groups, we can set the stage for the next step, which involves extracting these common factors. This step is not just about following a mechanical process; it requires a keen eye for patterns and relationships within the expression. The ability to effectively group terms is a hallmark of algebraic proficiency and is essential for mastering factoring techniques.

3. Identifying Common Factors

Now, we identify the greatest common factor (GCF) in each group.

• In the first group, $(6x^2 - 8x)$, the GCF is $2x$. We can factor out $2x$:

  $$2x(3x - 4)$$

• In the second group, $(21x - 28)$, the GCF is $7$. We can factor out $7$:

  $$7(3x - 4)$$

Identifying common factors is a pivotal step in the factoring process. The greatest common factor (GCF) represents the largest factor that divides evenly into all terms within a group. Recognizing and extracting the GCF simplifies the expression, making subsequent factoring steps more manageable. This process involves examining the coefficients and variables in each term to determine the highest number and variable powers that are shared among them. For instance, in the group $6x^2 - 8x$, both terms are divisible by 2, and they also share a common variable factor of $x$. Therefore, the GCF is $2x$. Similarly, in the group $21x - 28$, both terms are divisible by 7, making 7 the GCF. Extracting these common factors reveals the underlying structure of the expression and sets the stage for further factorization. Mastering the skill of identifying common factors is crucial for success in algebra and beyond.

4. Factoring out the Common Binomial

Notice that both groups now have a common binomial factor, $(3x - 4)$. We can factor this out:

$$2x(3x - 4) + 7(3x - 4) = (2x + 7)(3x - 4)$$

Factoring out the common binomial is a critical step in the factoring by grouping method. After identifying and extracting the greatest common factor (GCF) from each group of terms, a common binomial factor often emerges. This binomial factor is shared by both groups, allowing us to further simplify the expression. The process involves recognizing this shared binomial and factoring it out, effectively treating it as a single term. For example, if we have an expression like $2x(3x - 4) + 7(3x - 4)$, the common binomial factor is $(3x - 4)$. We can factor this out by rewriting the expression as $(2x + 7)(3x - 4)$. This step consolidates the expression into a product of two factors, representing the final factored form. The ability to identify and factor out common binomials is a key skill in algebra and is essential for solving equations and simplifying expressions.

5. Final Factored Expression

The expression is now completely factored:

$$6x^2 - 8x + 21x - 28 = (2x + 7)(3x - 4)$$

The final factored expression represents the culmination of the factoring process. It is the simplified form of the original expression, expressed as a product of its factors. In the case of the expression $6x^2 - 8x + 21x - 28$, the final factored form is $(2x + 7)(3x - 4)$. This means that if we were to multiply these two binomials together, we would obtain the original expression. The final factored expression is not just an end result; it is a valuable form for solving equations, simplifying expressions, and understanding the behavior of mathematical functions. It allows us to identify the roots or zeros of the expression, which are the values of $x$ that make the expression equal to zero. Additionally, the factored form can reveal important information about the graph of the function represented by the expression. Thus, the final factored expression is a powerful tool in algebra and calculus.

Alternative Method: Rearranging Terms

Sometimes, rearranging the terms before grouping can make the factoring process easier. Let’s try rearranging the terms in the original expression:

$$6x^2 + 21x - 8x - 28$$

1. Grouping Terms (Rearranged)

Now, we group the terms:

$$(6x^2 + 21x) + (-8x - 28)$$

Rearranging terms before grouping can be a strategic move in the factoring process. Sometimes, the initial arrangement of terms may not immediately reveal a clear path to factorization. By rearranging the terms, we can create groups that share common factors more readily. This technique involves carefully considering the coefficients and variables in the expression and looking for combinations that facilitate the identification of greatest common factors (GCFs). For instance, in the expression $6x^2 - 8x + 21x - 28$, rearranging the terms to $6x^2 + 21x - 8x - 28$ might make it easier to spot common factors within the groups. The goal is to create a grouping that allows for efficient extraction of common factors, leading to a simplified and factored expression. This step requires a flexible approach and a willingness to explore different arrangements to find the most effective path to factorization.

2. Identifying Common Factors (Rearranged)

• In the first group, $(6x^2 + 21x)$, the GCF is $3x$. Factoring out $3x$:

  $$3x(2x + 7)$$

• In the second group, $(-8x - 28)$, the GCF is $-4$. Factoring out $-4$:

  $$-4(2x + 7)$$

The process of identifying common factors in rearranged groups is similar to the initial method, but the rearrangement might make the common factors more apparent. After rearranging terms, we again look for the greatest common factor (GCF) within each group. This involves examining the coefficients and variables to find the largest factor that divides evenly into all terms within the group. For example, in the group $6x^2 + 21x$, the GCF is $3x$, as both terms are divisible by 3 and share a common variable factor of $x$. Similarly, in the group $-8x - 28$, the GCF is $-4$, as both terms are divisible by -4. Extracting these common factors simplifies the expression and prepares it for the next step, which involves factoring out the common binomial factor. The ability to identify common factors in rearranged groups is a valuable skill in factoring and can often lead to a more straightforward solution.

3. Factoring out the Common Binomial (Rearranged)

We see the common binomial factor $(2x + 7)$. Factoring it out:

$$3x(2x + 7) - 4(2x + 7) = (3x - 4)(2x + 7)$$

Factoring out the common binomial in the rearranged expression follows the same principle as in the initial method. After extracting the greatest common factor (GCF) from each group, we look for a binomial factor that is shared by both groups. This common binomial factor allows us to further simplify the expression by factoring it out. For instance, if we have an expression like $3x(2x + 7) - 4(2x + 7)$, the common binomial factor is $(2x + 7)$. We can factor this out by rewriting the expression as $(3x - 4)(2x + 7)$. This step consolidates the expression into a product of two factors, representing the final factored form. The ability to identify and factor out common binomials is a crucial skill in algebra and is essential for solving equations and simplifying expressions. This step demonstrates the power of rearranging terms to reveal hidden structures and facilitate factorization.

4. Final Factored Expression (Rearranged)

As we can see, the final factored expression is the same, just with the factors in a different order:

$$6x^2 + 21x - 8x - 28 = (3x - 4)(2x + 7)$$

The final factored expression, obtained after rearranging terms and factoring by grouping, confirms the consistency of the factoring process. Regardless of the initial arrangement of terms, the final factored form should be equivalent. In this case, the expression $6x^2 + 21x - 8x - 28$ factors to $(3x - 4)(2x + 7)$, which is the same as $(2x + 7)(3x - 4)$, just with the factors in a different order. This demonstrates that the order of factors does not affect the mathematical equivalence of the expression. The final factored expression is a valuable form for solving equations, simplifying expressions, and understanding the behavior of mathematical functions. It allows us to identify the roots or zeros of the expression and provides insights into the graph of the function. The consistency of the final factored form, regardless of the method used, underscores the fundamental principles of algebra.

Conclusion

In this comprehensive guide, we have successfully factored the expression $6x^2 - 8x + 21x - 28$ using factoring by grouping. We explored two approaches: the initial grouping and a method involving rearranging terms. Both methods led to the same factored form, $(2x + 7)(3x - 4)$, demonstrating the flexibility and power of factoring techniques. Mastering factoring is essential for success in algebra and beyond, as it forms the basis for solving equations, simplifying expressions, and tackling more complex mathematical problems.

By understanding the principles of factoring and practicing various methods, you can confidently approach a wide range of algebraic expressions and unlock their underlying structure. Factoring is not just a mathematical procedure; it's a problem-solving skill that enhances your analytical thinking and mathematical intuition. So, continue to practice and explore different factoring techniques to strengthen your algebraic prowess.