Factoring Expressions: Find The Factor Of 54xy + 45x - 18y - 15

Hey guys! Today, we're diving into a fun math problem focused on factoring algebraic expressions. Specifically, we're going to figure out which of the given options is a factor of the expression $54xy + 45x - 18y - 15$. Factoring can seem tricky at first, but with a systematic approach, it becomes much easier. So, let's break it down step by step.

Understanding the Problem

First, let’s restate the problem: We need to identify a factor of the expression $54xy + 45x - 18y - 15$. The options provided are:

• A. $6y + 5$
• B. $3y + 5$
• C. $6y + 1$
• D. $x - 5$

Our goal is to determine which of these expressions divides evenly into the original expression. To do this effectively, we'll employ the method of factoring by grouping. This technique involves grouping terms, factoring out common factors, and then identifying the shared binomial factor.

Factoring by grouping is a powerful technique that simplifies complex expressions into manageable parts. This approach is particularly useful when dealing with polynomials that have four or more terms. The underlying principle is to reorganize and factor subsets of the expression to reveal a common binomial factor, which can then be factored out to simplify the expression further. Mastering this technique not only helps in solving specific problems like the one at hand but also builds a solid foundation for more advanced algebraic manipulations. For instance, it is frequently used in solving quadratic equations and simplifying rational expressions. By practicing factoring by grouping, you enhance your ability to see patterns and relationships within algebraic expressions, making you a more confident and proficient problem solver.

Step-by-Step Solution

1. Grouping Terms

The first step in factoring by grouping is to pair the terms in the expression. A strategic grouping can reveal common factors that simplify the expression. We group the first two terms and the last two terms:

$(54xy + 45x) + (-18y - 15)$

This grouping sets the stage for the next step, where we will factor out the greatest common factor (GCF) from each group.

2. Factoring out the GCF

Now, let's identify and factor out the greatest common factor (GCF) from each group. For the first group, $(54xy + 45x)$, the GCF is $9x$. Factoring this out, we get:

$9x(6y + 5)$

For the second group, $(-18y - 15)$, the GCF is $-3$. Factoring this out, we get:

$-3(6y + 5)$

Now, rewrite the expression with the factored groups:

$9x(6y + 5) - 3(6y + 5)$

3. Identifying the Common Binomial Factor

Notice that both terms now share a common binomial factor: $(6y + 5)$. This is the key to completing the factorization. We can factor out this common binomial factor from the entire expression.

4. Factoring out the Binomial

Factor out the common binomial factor $(6y + 5)$:

$(6y + 5)(9x - 3)$

Now we have factored the expression into two factors. However, we can simplify further by factoring out the GCF from the second binomial factor.

5. Further Simplification

Look at the second factor, $(9x - 3)$. The greatest common factor here is 3. Factoring this out, we get:

$3(3x - 1)$

So, the fully factored expression is:

$(6y + 5) imes 3(3x - 1)$

Which can also be written as:

$3(6y + 5)(3x - 1)$

6. Identifying the Correct Answer

Now, let's compare our factored expression with the given options. We are looking for a factor of the original expression.

Our factored form includes $(6y + 5)$ as one of the factors. Comparing this with the given options:

• A. $6y + 5$ - This matches one of our factors.
• B. $3y + 5$ - This does not match.
• C. $6y + 1$ - This does not match.
• D. $x - 5$ - This does not match.

Therefore, the correct answer is A. $6y + 5$.

Why This Method Works

Factoring by grouping works because it leverages the distributive property in reverse. By strategically grouping terms and factoring out common factors, we create a situation where a binomial factor becomes apparent. This shared binomial allows us to further simplify the expression, effectively breaking it down into its constituent parts. The beauty of this method lies in its ability to transform a seemingly complex polynomial into a product of simpler expressions, making it easier to analyze and manipulate.

Understanding the underlying principles of factoring, such as the distributive property, enhances your ability to apply these techniques effectively. It’s not just about memorizing steps but comprehending why these steps lead to the correct solution. This deeper understanding makes you a more versatile and confident mathematician, capable of tackling a wider range of problems.

Common Mistakes to Avoid

When factoring by grouping, it’s easy to make a few common mistakes. Let’s go over these so you can steer clear of them:

• Incorrectly Factoring out the GCF: Make sure you’re pulling out the greatest common factor. Sometimes it’s tempting to take out a smaller factor, but this leaves more work for later and can complicate the process.
• Sign Errors: Pay close attention to signs, especially when factoring out a negative number. A mistake here can throw off the whole problem.
• Forgetting to Distribute: Remember to distribute the GCF back into the parentheses to check your work. This ensures you haven’t changed the value of the expression.
• Stopping Too Early: Sometimes, you might find a common factor but not simplify completely. Always look for further factoring opportunities.

Avoiding these mistakes comes down to careful practice and attention to detail. Always double-check your work and take your time to ensure accuracy.

Practice Problems

To really nail factoring by grouping, practice is key! Here are a few problems you can try on your own:

1. $15xy + 10x + 9y + 6$
2. $21pq - 6p + 35q - 10$
3. $12ab - 18a - 10b + 15$

Work through these problems step by step, applying the techniques we’ve discussed. Check your answers with online resources or ask a friend for help. The more you practice, the more comfortable you’ll become with factoring.

Conclusion

Alright guys, we’ve successfully identified that $6y + 5$ is a factor of the expression $54xy + 45x - 18y - 15$. We achieved this by using the method of factoring by grouping, which involves grouping terms, factoring out common factors, and simplifying. Remember, practice is crucial for mastering these techniques. Keep working at it, and you’ll become a factoring pro in no time! Happy factoring!