Factoring Out 2: Step-by-Step Solutions & Examples

Hey guys! Today, we're diving into the world of factoring, specifically how to factor out the number 2 from various expressions. Factoring might sound intimidating, but trust me, it's like finding the hidden pieces of a puzzle. Once you get the hang of it, it becomes super useful in simplifying equations and solving problems. So, let's break it down and make it easy to understand. We will go through five different examples, each with its unique twist, to ensure you grasp the concept completely. Buckle up, and let’s get started!

Understanding Factoring and the Common Factor

Before we jump into the examples, let’s quickly recap what factoring is all about. In simple terms, factoring is like the reverse of expanding. When we expand, we multiply a term across a bracket. Factoring, on the other hand, is finding what common factors we can pull out of an expression to rewrite it in a more simplified form. Think of it like this: instead of building something up through multiplication, we're breaking it down into its multiplicative components.

The common factor is the key player here. It's a number or variable that divides evenly into all terms in the expression. In our case today, we’re focusing on the number 2 as the common factor. This means we're looking for terms that are multiples of 2. Identifying this common factor is the first and most crucial step in the factoring process. Once you spot it, the rest is just a matter of rewriting the expression.

Why is factoring so important, you might ask? Well, it's a fundamental skill in algebra and higher mathematics. Factoring helps in simplifying complex expressions, solving equations, and understanding the structure of mathematical relationships. It's like having a secret weapon in your math toolkit! Now that we've got the basics down, let’s move on to the fun part: tackling those examples.

a) Factoring 2 from 2a + 2b

Our first expression is 2a + 2b. The goal here is to identify the common factor in both terms. Looking at it, we can see that both terms, 2a and 2b, have the number 2 as a common factor. This is pretty straightforward, right? The number 2 is multiplying both a and b.

So, how do we factor it out? We simply take the common factor, which is 2, and write it outside a set of parentheses. Inside the parentheses, we write what’s left of each term after we've divided out the 2. For the term 2a, if we divide out the 2, we are left with a. Similarly, for the term 2b, dividing out the 2 leaves us with b. So, we put a and b inside the parentheses, connected by the same operation that was in the original expression, which is addition in this case.

Therefore, the factored form of 2a + 2b is 2(a + b). It's as simple as that! We've effectively rewritten the expression by pulling out the common factor. To check our work, we can always distribute the 2 back into the parentheses. If we do that, we get 2 * a + 2 * b, which simplifies to 2a + 2b—our original expression. This confirms that our factoring is correct.

Factoring out common factors like this is a foundational skill in algebra. It allows us to simplify expressions and is a crucial step in solving more complex equations. By identifying and extracting the common factor, we make the expression easier to work with. Now, let’s move on to our next example and see how this works in a slightly different context.

b) Factoring 2 from 2 · 35 + 2 · 65

Next up, we have the expression 2 · 35 + 2 · 65. At first glance, this might look a bit different from our previous example, but the principle remains the same. We are still on the lookout for the common factor, and in this case, it’s quite evident: the number 2 appears in both terms.

The first term is 2 · 35, which means 2 multiplied by 35, and the second term is 2 · 65, which is 2 multiplied by 65. Both terms clearly have 2 as a factor. So, just like before, we will factor out the 2. We write the 2 outside a set of parentheses, and then we need to figure out what goes inside.

To determine what goes inside the parentheses, we consider what’s left after we divide each term by the common factor, which is 2. When we divide 2 · 35 by 2, we are left with 35. Similarly, when we divide 2 · 65 by 2, we are left with 65. So, inside the parentheses, we will have 35 and 65, connected by the same operation as in the original expression, which is addition.

Therefore, the factored form of 2 · 35 + 2 · 65 is 2(35 + 65). We have successfully factored out the 2. Now, if we want, we can further simplify the expression inside the parentheses. 35 plus 65 equals 100. So, the expression becomes 2(100), which simplifies to 200. This shows how factoring can be a stepping stone to simplifying expressions and arriving at a numerical answer.

This example illustrates that factoring isn't just about variables; it applies to numerical expressions as well. Recognizing the common factor helps us rewrite the expression in a more manageable form. Let’s move on to our next example, where we’ll tackle a slightly different situation.

c) Factoring 2 from 2 + 22z

Now, let's tackle the expression 2 + 22z. In this case, we still need to identify the common factor, and once again, it's the number 2. The first term is simply 2, and the second term is 22z.

It's clear that 2 is a factor of the first term (since 2 divided by 2 is 1). But is 2 also a factor of 22z? Absolutely! The number 22 is a multiple of 2 (22 = 2 * 11). So, we can confidently say that 2 is a common factor in both terms. Now, let’s factor it out.

We write the 2 outside the parentheses. Inside the parentheses, we need to write what’s left after we divide each term by 2. For the first term, 2 divided by 2 is 1. It’s crucial to remember to include this 1 inside the parentheses. For the second term, 22z divided by 2 is 11z (since 22 / 2 = 11). So, we have 11z as the second term inside the parentheses.

Thus, the factored form of 2 + 22z is 2(1 + 11z). We’ve successfully factored out the 2. It's worth noting that we can't simplify this expression further unless we have a specific value for z. However, by factoring out the common factor, we've rewritten the expression in a more simplified form, which can be very useful in various algebraic manipulations.

This example highlights the importance of remembering the 1 when factoring out a term completely. It's a common mistake to overlook this, but including the 1 ensures that the factored form is equivalent to the original expression. Let’s move on to the next example, where we’ll explore a scenario involving subtraction.

d) Factoring 2 from 8m - 2

Our next expression is 8m - 2. In this case, we're looking to factor out the number 2, and it’s quite doable. The first term is 8m, and the second term is -2. Notice the subtraction sign; it's important to keep track of that as we factor.

First, let's consider 8m. Is 2 a factor of 8m? Yes, it is, because 8 is a multiple of 2 (8 = 2 * 4). So, we can divide 8m by 2. Now, let’s look at the second term, -2. Is 2 a factor of -2? Yes, it is, since -2 divided by 2 is -1.

Now that we’ve confirmed that 2 is a common factor, let’s factor it out. We write the 2 outside the parentheses. Inside the parentheses, we write what’s left after we divide each term by 2. For the first term, 8m divided by 2 is 4m. For the second term, -2 divided by 2 is -1. It’s crucial to carry the negative sign along with the 1.

So, the factored form of 8m - 2 is 2(4m - 1). We’ve successfully factored out the 2, and we’ve made sure to include the negative sign in the second term inside the parentheses. This example demonstrates that factoring works seamlessly with subtraction as well.

This example also reinforces the importance of paying attention to signs. When factoring, it's essential to ensure that the signs inside the parentheses are correct to maintain the equivalence of the expressions. Now, let’s move on to our final example, which is a bit more complex and involves a series of numbers.

e) Factoring 2 from 2 + 4 + 6 + … + 100

Finally, we have the expression 2 + 4 + 6 + … + 100. This one looks a bit different because it's a series of numbers, but the principle of factoring remains the same. We need to identify the common factor, and in this case, it’s quite clear: every number in this series is a multiple of 2.

The series consists of even numbers from 2 up to 100. This means we can factor out the number 2 from each term. So, let’s do that. We write the 2 outside the parentheses. Inside the parentheses, we need to write what’s left after we divide each term by 2.

When we divide 2 by 2, we get 1. When we divide 4 by 2, we get 2. When we divide 6 by 2, we get 3. And so on. This pattern continues until we reach the last term. When we divide 100 by 2, we get 50. So, inside the parentheses, we will have the series 1 + 2 + 3 + … + 50.

Therefore, the factored form of 2 + 4 + 6 + … + 100 is 2(1 + 2 + 3 + … + 50). We’ve successfully factored out the 2. Now, we might be curious about what the sum of the numbers inside the parentheses is. This is a classic arithmetic series, and there’s a formula to calculate it quickly. The sum of the first n natural numbers is given by n(n + 1) / 2.

In our case, n is 50, so the sum of the numbers from 1 to 50 is 50 * (50 + 1) / 2, which is 50 * 51 / 2. This simplifies to 2550 / 2, which equals 1275. So, the sum inside the parentheses is 1275. Therefore, the entire expression simplifies to 2 * 1275, which equals 2550. This example not only demonstrates factoring but also shows how it can be combined with other mathematical concepts to solve problems.

This final example illustrates that factoring can be applied to a wide range of expressions, including series of numbers. By identifying the common factor, we can rewrite the expression in a more simplified form and potentially make further calculations easier.

Conclusion

Alright guys, we’ve reached the end of our factoring journey for today! We've tackled five different examples, each showing how to factor out the number 2 from various expressions. From simple algebraic terms to numerical series, we've seen that the principle remains consistent: identify the common factor and rewrite the expression by pulling it out.

Factoring is a fundamental skill in mathematics, and mastering it opens the door to more advanced concepts. It’s like learning a new language – once you understand the grammar, you can express yourself in countless ways. In math, factoring helps you simplify expressions, solve equations, and understand mathematical relationships more deeply.

Remember, practice makes perfect. The more you work with factoring, the more natural it will become. So, keep practicing, keep exploring, and don't hesitate to tackle new challenges. You've got this! And who knows, maybe next time, we'll explore factoring out other numbers or even variables. Until then, keep factoring and keep learning!