Factoring Polynomials Rewrite 6x^2 + 2xy + 9x + 3y In Factored Form

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Hey guys! Today, we're going to break down the expression $6x^2 + 2xy + 9x + 3y$ and rewrite it in its factored form. Factoring can sometimes seem like a puzzle, but with the right approach, it becomes a lot easier. We’ll go through the steps together, so you can see exactly how it’s done. Our goal is to identify common factors and group terms in a way that simplifies the expression. So, let’s dive in and get started!

Understanding Factoring

Before we jump into the problem, let's quickly recap what factoring means. Factoring is essentially the reverse of expanding. When we expand, we multiply terms together, like distributing a number across a parenthesis. Factoring, on the other hand, involves breaking down an expression into the product of its factors. Think of it like finding the ingredients that, when multiplied together, give us the original expression.

In our case, we have the expression $6x^2 + 2xy + 9x + 3y$. Our mission is to find two binomials (expressions with two terms) that, when multiplied, will give us this expression. There are several techniques we can use, but for this particular expression, we'll use a method called factoring by grouping. This method is super useful when you have four terms, like we do here.

Keywords in factoring include greatest common factor, grouping, and binomials. Factoring by grouping is particularly effective when you notice terms that share common factors, which can be grouped together to simplify the expression. For instance, in our expression, $6x^2$ and $2xy$ share common factors, and $9x$ and $3y$ also share common factors. Identifying these commonalities is the first step in our factoring journey.

Step-by-Step Factoring Process

Alright, let's get into the nitty-gritty of factoring $6x^2 + 2xy + 9x + 3y$. We'll break it down step by step so you can follow along easily.

Step 1: Grouping Terms

The first thing we want to do is group the terms in a way that makes sense. Look for pairs of terms that have a common factor. In our expression, $6x^2$ and $2xy$ have common factors, and so do $9x$ and $3y$. Let’s group them together:

$(6x^2 + 2xy) + (9x + 3y)$

Grouping terms like this is a crucial step because it sets us up to factor out common factors from each group. It’s like organizing your puzzle pieces before you start assembling the puzzle. By grouping, we're creating smaller, more manageable chunks that we can work with.

Step 2: Factoring Out Common Factors

Now that we've grouped our terms, we need to factor out the greatest common factor (GCF) from each group. The GCF is the largest factor that divides evenly into all terms in the group. For the first group, $(6x^2 + 2xy)$, the GCF is $2x$. For the second group, $(9x + 3y)$, the GCF is $3$. Let's factor these out:

$2x(3x + y) + 3(3x + y)$

Notice something cool here? We now have two terms, and both of them have a common factor of $(3x + y)$. This is exactly what we want! Factoring out common factors simplifies the expression and reveals the underlying structure.

Step 3: Factoring Out the Common Binomial

We’ve made some great progress! Now, we have $2x(3x + y) + 3(3x + y)$. As we noticed, both terms have a common binomial factor of $(3x + y)$. We can factor this out just like we factored out the GCF in the previous step:

$(3x + y)(2x + 3)$

And there you have it! We've successfully factored the expression $6x^2 + 2xy + 9x + 3y$ into $(3x + y)(2x + 3)$. This is our factored form. Factoring out the common binomial is the final step in our journey, bringing us to a neat and simplified expression.

Verifying the Factored Form

It's always a good idea to double-check your work, especially in math. We can verify our factored form by expanding it and making sure it matches the original expression. Let's expand $(3x + y)(2x + 3)$ using the FOIL method (First, Outer, Inner, Last):

• First: $(3x)(2x) = 6x^2$
• Outer: $(3x)(3) = 9x$
• Inner: $(y)(2x) = 2xy$
• Last: $(y)(3) = 3y$

Now, let’s add these terms together:

$6x^2 + 9x + 2xy + 3y$

Rearranging the terms, we get:

$6x^2 + 2xy + 9x + 3y$

This matches our original expression! So, we know our factored form is correct. Verifying your answer is a fantastic way to ensure accuracy and build confidence in your factoring skills. It’s like having a secret code to check your solution!

Conclusion: Putting It All Together

So, we started with the expression $6x^2 + 2xy + 9x + 3y$ and, through the magic of factoring by grouping, we transformed it into $(3x + y)(2x + 3)$. Remember, guys, the key steps were:

1. Grouping terms with common factors.
2. Factoring out the greatest common factor from each group.
3. Factoring out the common binomial.

Factoring might seem intimidating at first, but with practice, it becomes second nature. By breaking down the expression into manageable steps, we were able to tackle it with ease. Always remember to verify your answer to ensure accuracy. Factoring is a powerful tool in algebra, and mastering it will open doors to more advanced mathematical concepts.

Keywords like factoring by grouping, binomial factors, and greatest common factors are essential for understanding and applying this technique. Keep practicing, and you'll become a factoring pro in no time! Now you know how to rewrite $6x^2 + 2xy + 9x + 3y$ in factored form.

Okay, let’s take a look at the answer choices provided and see which one matches our factored form. This is a crucial step in any math problem, as it helps us solidify our understanding and ensure we’ve arrived at the correct solution. We'll go through each option, comparing it to our result and explaining why it is or isn't the correct answer.

Evaluating the Options

We’ve determined that the factored form of $6x^2 + 2xy + 9x + 3y$ is $(3x + y)(2x + 3)$. Now, let's examine the answer choices:

A. $(2x + 3)(3x + y)$ B. $(2x + y)(3x + 3)$ C. $(3x + 2)(3x + y)$ D. $(3x + y)(2x + 1)$

Option A: $(2x + 3)(3x + y)$

This looks awfully familiar, doesn't it? In fact, it's exactly what we got! Remember, multiplication is commutative, meaning the order in which we multiply factors doesn't change the result. So, $(3x + y)(2x + 3)$ is the same as $(2x + 3)(3x + y)$. This option matches our factored form perfectly.

Option B: $(2x + y)(3x + 3)$

Let’s expand this to see if it matches our original expression:

$(2x + y)(3x + 3) = 2x(3x) + 2x(3) + y(3x) + y(3) = 6x^2 + 6x + 3xy + 3y$

This does not match our original expression $6x^2 + 2xy + 9x + 3y$. The terms $6x$ and $3xy$ are different from $9x$ and $2xy$, so this option is incorrect.

Option C: $(3x + 2)(3x + y)$

Let’s expand this one too:

$(3x + 2)(3x + y) = 3x(3x) + 3x(y) + 2(3x) + 2(y) = 9x^2 + 3xy + 6x + 2y$

Again, this does not match our original expression. The presence of $9x^2$ and the differing coefficients make it clear that this is not the correct factored form.

Option D: $(3x + y)(2x + 1)$

And let’s expand the last option:

$(3x + y)(2x + 1) = 3x(2x) + 3x(1) + y(2x) + y(1) = 6x^2 + 3x + 2xy + y$

This also does not match our original expression. The terms $3x$ and $y$ are different from $9x$ and $3y$, indicating that this option is incorrect.

Conclusion of Answer Choices

After carefully analyzing each option, it's clear that Option A, $(2x + 3)(3x + y)$, is the correct factored form of the expression $6x^2 + 2xy + 9x + 3y$. We arrived at this conclusion by factoring the expression ourselves and then verifying it by expanding the factors. We also took the time to expand the incorrect options to clearly see why they didn't match the original expression. This thorough approach ensures we not only find the right answer but also understand why the other options are wrong.

By understanding how to evaluate each choice, you’re not just guessing; you’re applying your knowledge of factoring and expansion to arrive at the correct answer. This skill is invaluable in algebra and beyond, guys!

Keywords such as expanding factored forms, verifying solutions, and matching expressions are crucial when analyzing answer choices in factoring problems. Always take the time to check your work and understand why each option is correct or incorrect.

Alright, let’s talk strategy! Factoring can feel like a maze sometimes, but with the right approaches, you can navigate it like a pro. We'll cover some key strategies that will help you tackle factoring problems with confidence. These strategies aren’t just for this specific problem; they’re tools you can use in a wide range of algebraic situations. Whether you're dealing with quadratics, polynomials, or any other expression that needs factoring, these tips will be your best friends.

Essential Factoring Techniques

1. Look for the Greatest Common Factor (GCF)

This is the golden rule of factoring! Always, always, always start by looking for the greatest common factor (GCF). It’s the largest factor that divides evenly into all terms in the expression. Factoring out the GCF simplifies the expression and makes it easier to work with. Think of it as decluttering before you start organizing. In our example, if we hadn't grouped terms and factored, the GCF wouldn't have been as obvious.

2. Factoring by Grouping

We used this technique to solve our main problem, and it’s super handy when you have four terms. The key is to group terms that share common factors, then factor out those common factors. If you’ve done it right, you’ll end up with a common binomial factor that you can factor out again. Remember, this strategy is all about creating manageable chunks within the expression.

3. Recognize Special Patterns

There are certain patterns that pop up frequently in factoring, and recognizing them can save you a ton of time. These include:

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$
• Perfect Square Trinomials: $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$

Keeping an eye out for these patterns is like having a shortcut in your factoring toolbox. It allows you to quickly identify factors without going through the full grouping process.

4. Trial and Error (with Strategy)

Sometimes, factoring requires a bit of trial and error, especially with quadratic expressions. But don’t just guess randomly! Use a systematic approach. Think about the factors of the leading coefficient and the constant term, and try different combinations until you find the one that works. This isn't blind guessing; it's educated trial and error.

General Problem-Solving Tips

1. Read the Problem Carefully

This might seem obvious, but it’s crucial. Make sure you understand exactly what the problem is asking. Are you supposed to factor completely? Are there any specific instructions? Missing a small detail can lead you down the wrong path.

2. Show Your Work

Showing your steps is not just for your teacher; it’s for you! Writing out each step makes it easier to track your progress, identify mistakes, and understand the process. Plus, if you do make a mistake, you’ll be able to see where you went wrong and learn from it.

3. Check Your Answer

We talked about this earlier, but it’s worth repeating. Always check your answer! Expand your factored form to make sure it matches the original expression. This simple step can save you from making careless errors and build your confidence in your solutions.

4. Practice, Practice, Practice!

Like any skill, factoring gets easier with practice. The more problems you solve, the more comfortable you’ll become with different techniques and patterns. So, grab some practice problems and get to work!

Bringing It All Together

By using these strategies, you’ll be well-equipped to tackle a wide range of factoring problems. Remember, start by looking for the GCF, consider factoring by grouping, recognize special patterns, and don’t be afraid to use trial and error strategically. And always, always, always check your answer! These tips, combined with consistent practice, will turn you into a factoring whiz, guys!

Keywords such as GCF, factoring patterns, trial and error, and problem-solving strategies are essential for mastering factoring. Keep these techniques in mind, and you'll be able to approach any factoring problem with confidence.

Alright, let’s talk about some common pitfalls in factoring. We all make mistakes, but knowing what to watch out for can help you avoid them. We'll cover some typical errors that students make when factoring and give you tips on how to dodge these traps. Recognizing these mistakes is like having a map of the danger zones in the factoring landscape. Armed with this knowledge, you’ll be able to steer clear of them and factor like a pro.

Typical Factoring Errors

1. Forgetting to Factor Out the GCF

We’ve said it before, but it’s worth repeating: Always start by looking for the greatest common factor (GCF). Forgetting this step is a super common mistake, and it can lead to incorrect factoring. Factoring out the GCF first simplifies the expression and makes subsequent steps easier. It's like laying a solid foundation before building a house.

How to Avoid It: Make it a habit to always check for a GCF before doing anything else. Ask yourself,