Factoring 2ax² + 6ax² Step-by-Step Solution

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Hey guys! Today, we're diving deep into the world of factoring, specifically focusing on the expression 2ax² + 6ax². Factoring might seem like a daunting task at first, but trust me, with a little understanding and practice, it becomes a piece of cake. We'll break down the steps, explain the concepts, and make sure you're confident in tackling similar problems. So, let's get started!

Understanding Factoring: The Basics

Before we jump into the expression 2ax² + 6ax², let's quickly recap what factoring actually means. In simple terms, factoring is like reverse multiplication. Think of it as breaking down a number or an expression into its constituent parts – the things that were multiplied together to get the original thing. For example, the number 12 can be factored into 2 x 2 x 3. Similarly, an algebraic expression can be factored into simpler expressions.

In the context of algebra, factoring involves identifying common factors within an expression and extracting them. This process simplifies the expression and can be super useful for solving equations, simplifying fractions, and a bunch of other cool stuff in math. There are several techniques for factoring, including:

  • Greatest Common Factor (GCF): This is the most basic and often the first method you should try. It involves finding the largest factor that divides all terms in the expression.
  • Difference of Squares: This applies to expressions in the form a² - b².
  • Perfect Square Trinomials: These are trinomials that can be expressed as (ax + b)² or (ax - b)².
  • Factoring by Grouping: This technique is used for expressions with four or more terms.
  • Factoring Quadratics: This involves factoring expressions in the form ax² + bx + c.

For our expression, 2ax² + 6ax², we'll primarily use the Greatest Common Factor (GCF) method, as it's the most straightforward approach in this case. Understanding the GCF is crucial for simplifying expressions effectively, and it forms the foundation for more advanced factoring techniques. So, let's dive into how we can apply this to our expression.

Identifying the Greatest Common Factor (GCF) in 2ax² + 6ax²

The heart of factoring lies in identifying the Greatest Common Factor (GCF). The GCF, as the name suggests, is the largest factor that is common to all terms in an expression. To find the GCF in 2ax² + 6ax², we need to break down each term and see what factors they share. Let's do this step-by-step.

First, let's look at the coefficients, which are the numerical parts of the terms. In our expression, we have 2 and 6. The factors of 2 are 1 and 2, while the factors of 6 are 1, 2, 3, and 6. The largest number that appears in both lists is 2, so the numerical part of our GCF is 2. Easy peasy, right?

Next, we'll examine the variables. Both terms in our expression, 2ax² and 6ax², contain the variables a and x. In the first term, a appears with a power of 1 (which we usually don't write explicitly), and x appears with a power of 2. In the second term, a also appears with a power of 1, and x appears with a power of 2. So, both terms share a and x². When identifying the GCF for variables, we take the lowest power of the common variables present in all terms. In this case, the lowest power of a is 1, and the lowest power of x is 2. Therefore, the variable part of our GCF is ax².

Now, we combine the numerical and variable parts of the GCF. We found that the numerical GCF is 2 and the variable GCF is ax². Putting them together, the GCF of 2ax² + 6ax² is 2ax². Identifying the GCF is a critical step because it allows us to simplify the expression and make it easier to work with. By systematically breaking down each term and comparing their factors, we can confidently find the largest common factor. Now that we've found the GCF, we're ready to move on to the actual factoring process. Let's see how we can use this GCF to rewrite our expression in a factored form.

Factoring Out the GCF from 2ax² + 6ax²

Alright, we've successfully identified the GCF of our expression 2ax² + 6ax² as 2ax². Now comes the exciting part – actually factoring it out! Factoring out the GCF is like reverse distribution. We're essentially dividing each term in the expression by the GCF and then writing the expression as a product of the GCF and the resulting quotient.

Let's break it down step-by-step. First, we'll write down our GCF, which is 2ax². Then, we'll open a set of parentheses, which will contain the result of dividing each term in the original expression by the GCF. So, we have:

2ax²( )

Now, we'll divide each term in 2ax² + 6ax² by 2ax². Let's start with the first term, 2ax². When we divide 2ax² by 2ax², we get 1. Remember, anything divided by itself is 1. So, we write 1 inside the parentheses:

2ax²(1 )

Next, we move on to the second term, 6ax². We'll divide 6ax² by 2ax². The coefficients 6 divided by 2 gives us 3. The variables ax² divided by ax² gives us 1 (they cancel each other out). So, the result is 3. We add this to the parentheses:

2ax²(1 + 3)

And that's it! We've successfully factored out the GCF. The factored form of 2ax² + 6ax² is 2ax²(1 + 3). This might seem like a small step, but it's a significant simplification of the original expression. Factoring out the GCF not only makes the expression more manageable but also reveals its underlying structure. It's like peeling back the layers to see what's truly inside. Now, we can take this a step further and simplify the expression inside the parentheses. Let's see what that looks like.

Simplifying the Factored Expression

Okay, so we've factored 2ax² + 6ax² into 2ax²(1 + 3). The next logical step is to simplify the expression inside the parentheses. This part is super straightforward – we just need to perform the addition.

Inside the parentheses, we have 1 + 3. Adding these two numbers together gives us 4. So, we can replace (1 + 3) with (4). Our expression now looks like this:

2ax²(4)

Now, we have a product of 2ax² and 4. To further simplify, we can multiply the coefficients together. The coefficient of the first term is 2, and the second term is 4. Multiplying 2 by 4 gives us 8. So, we replace 2ax²(4) with 8ax². Our fully simplified expression is:

8ax²

And there you have it! By factoring out the GCF and simplifying, we've transformed 2ax² + 6ax² into 8ax². This simplified form is much easier to work with in various mathematical contexts. It's a testament to the power of factoring – taking a seemingly complex expression and reducing it to its simplest form.

Simplifying the factored expression is a crucial step in the factoring process. It not only makes the expression more concise but also helps in identifying the true nature of the expression. In many cases, the simplified form reveals patterns and relationships that were not immediately apparent in the original expression. So, always remember to simplify after factoring, as it often leads to a more elegant and useful result.

Conclusion: The Result of Factoring 2ax² + 6ax²

So, let's recap our journey through factoring the expression 2ax² + 6ax². We started by understanding the basics of factoring and the importance of the Greatest Common Factor (GCF). We then identified the GCF of 2ax² + 6ax² as 2ax². After that, we factored out the GCF, which gave us 2ax²(1 + 3). Finally, we simplified the expression inside the parentheses and multiplied the coefficients, leading us to our final result:

8ax²

Therefore, when we factor the expression 2ax² + 6ax², we obtain 8ax² as the result. This exercise highlights the power of factoring in simplifying algebraic expressions. By identifying common factors and extracting them, we can transform complex expressions into simpler, more manageable forms.

Factoring is a fundamental skill in algebra and is used extensively in various areas of mathematics, including solving equations, simplifying fractions, and analyzing functions. Mastering factoring techniques not only helps in solving specific problems but also enhances your overall mathematical intuition and problem-solving abilities.

I hope this comprehensive guide has helped you understand how to factor the expression 2ax² + 6ax² and has given you a solid foundation in factoring techniques. Remember, practice makes perfect, so keep working on similar problems to build your confidence and skills. Keep up the great work, guys, and happy factoring!