Factoring 15x² + 3xy + 10x + 2y A Step-by-Step Guide

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In the realm of algebra, factorization stands as a fundamental technique for simplifying expressions and solving equations. When confronted with a polynomial like 15x² + 3xy + 10x + 2y, the task of identifying its factors might seem daunting at first. However, by employing strategic methods and a keen eye for patterns, we can systematically break down this expression into its constituent factors. This article will delve into a detailed exploration of the factorization process, providing a step-by-step guide to unravel the factors of the given polynomial.

Understanding Factorization

Before we embark on the journey of factoring 15x² + 3xy + 10x + 2y, it is crucial to grasp the essence of factorization itself. In essence, factorization is the reverse process of expansion. When we expand an expression, we multiply terms together to obtain a more complex expression. Conversely, when we factorize an expression, we decompose it into simpler expressions that, when multiplied together, yield the original expression.

Factorization serves as a powerful tool in various mathematical contexts. It allows us to simplify complex expressions, making them easier to manipulate and solve. Moreover, factorization plays a vital role in solving equations, particularly quadratic equations, where it helps us find the roots or solutions of the equation.

Strategies for Factorization

Several strategies can be employed to factorize algebraic expressions. Among the most common techniques are:

  • Common Factoring: This involves identifying the greatest common factor (GCF) among the terms of the expression and factoring it out.
  • Grouping: This technique is particularly useful when dealing with expressions containing four or more terms. It involves grouping terms together in pairs and then factoring out common factors from each pair.
  • Difference of Squares: This method applies to expressions in the form a² - b², which can be factored as (a + b)(a - b).
  • Perfect Square Trinomials: Trinomials in the form a² + 2ab + b² or a² - 2ab + b² can be factored as (a + b)² or (a - b)², respectively.
  • Trial and Error: This method involves systematically trying different combinations of factors until the correct factorization is found.

Factorizing 15x² + 3xy + 10x + 2y: A Step-by-Step Approach

Now, let's apply these strategies to factorize the expression 15x² + 3xy + 10x + 2y. Given the four terms in the expression, the grouping method appears to be the most promising approach. Our goal is to strategically group the terms in pairs and then factor out common factors from each pair.

Step 1: Grouping Terms

We can group the terms as follows:

(15x² + 3xy) + (10x + 2y)

Step 2: Factoring out Common Factors

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

In the first group (15x² + 3xy), the GCF is 3x. Factoring out 3x, we get:

3x(5x + y)

In the second group (10x + 2y), the GCF is 2. Factoring out 2, we get:

2(5x + y)

Step 3: Combining the Factored Terms

Now, we have:

3x(5x + y) + 2(5x + y)

Observe that the expression (5x + y) is common to both terms. We can factor out (5x + y) from the entire expression:

(5x + y)(3x + 2)

The Factors of 15x² + 3xy + 10x + 2y

Therefore, the factors of 15x² + 3xy + 10x + 2y are (5x + y) and (3x + 2). This corresponds to answer choice C. (3x + y)(5x + 2) in the original problem.

Verification: Expanding the Factors

To ensure that our factorization is correct, we can expand the factors (5x + y) and (3x + 2) and verify that we obtain the original expression 15x² + 3xy + 10x + 2y.

Expanding (5x + y)(3x + 2), we get:

(5x + y)(3x + 2) = 5x(3x + 2) + y(3x + 2)

= 15x² + 10x + 3xy + 2y

= 15x² + 3xy + 10x + 2y

As we can see, the expanded expression matches the original expression, confirming that our factorization is correct.

Alternative Grouping (Exploration)

While the grouping we chose initially led us to the solution, it's worth exploring if an alternative grouping might also work. Let's try grouping the terms differently:

(15x² + 10x) + (3xy + 2y)

Now, we factor out the GCF from each group:

5x(3x + 2) + y(3x + 2)

Notice that (3x + 2) is a common factor in both terms. Factoring it out, we get:

(3x + 2)(5x + y)

This is the same factorization we obtained earlier, just with the factors written in a different order. This demonstrates that sometimes different groupings can lead to the same result, reinforcing the flexibility of the grouping method.

Common Mistakes to Avoid

Factorization can be tricky, and it's easy to make mistakes if one isn't careful. Here are some common pitfalls to watch out for:

  • Incorrectly Identifying GCFs: Always ensure that you're factoring out the greatest common factor, not just any common factor. For example, in 15x² + 3xy, factoring out 3x is correct, but factoring out just 'x' would leave you with 15x + 3y inside the parentheses, which still has a common factor.
  • Sign Errors: Pay close attention to signs when factoring out negative numbers. For instance, if you had -10x - 2y, factoring out -2 should give you -2(5x + y).
  • Incomplete Factorization: Make sure you've factored the expression completely. After factoring, double-check if any of the resulting factors can be factored further.
  • Assuming a Pattern: Don't jump to conclusions about the type of factorization needed. Always start with the simplest methods like common factoring and grouping before trying more complex techniques.

Importance of Practice

Like any mathematical skill, factorization becomes easier with practice. The more you work through different types of problems, the better you'll become at recognizing patterns and applying the appropriate techniques. Start with simpler expressions and gradually work your way up to more complex ones. Online resources, textbooks, and practice worksheets can be invaluable tools in honing your factorization skills.

Conclusion

In conclusion, the factors of 15x² + 3xy + 10x + 2y are (5x + y) and (3x + 2). We arrived at this solution by employing the grouping method, a powerful technique for factorizing expressions with four or more terms. By strategically grouping terms, factoring out common factors, and recognizing patterns, we successfully decomposed the given polynomial into its constituent factors.

Mastering factorization is a crucial step in your algebraic journey. It not only simplifies expressions and solves equations but also lays the foundation for more advanced mathematical concepts. Remember to practice regularly, avoid common mistakes, and embrace the challenge of unraveling the hidden factors within algebraic expressions. With dedication and perseverance, you'll become a proficient factorizer and unlock new levels of mathematical understanding.