Factoring With Common Factors: Step-by-Step Examples

Hey guys! Today, we're diving into the world of factoring using common factors. It's a super useful skill in math that simplifies expressions and makes solving equations a whole lot easier. We'll break down a few examples step-by-step, so you can master this technique. Let's get started!

Understanding Common Factors

Before we jump into the examples, let's quickly recap what a common factor is. A common factor is a number or expression that divides evenly into two or more terms. Factoring, in general, is the process of breaking down an expression into its multiplicative components. When we talk about factoring using a common factor, we're looking for that shared element that can be pulled out to simplify the expression.

Why is this important? Well, factoring simplifies complex expressions, reveals hidden structures, and is crucial for solving equations. When you identify a common factor, you're essentially reversing the distributive property. This is a fundamental concept in algebra, so understanding it well is key to tackling more advanced topics. It helps in simplifying algebraic expressions, solving equations, and understanding relationships between different terms. Trust me, once you get the hang of it, you'll see common factors everywhere!

Example A: 3 * 45 + 3 * 15

Let's start with our first example: 3 * 45 + 3 * 15. The goal here is to identify the common factor in both terms. Looking at the expression, we can clearly see that the number 3 appears in both 3 * 45 and 3 * 15. This is our common factor.

To factor this expression, we pull out the common factor 3 and rewrite the expression using the distributive property in reverse. This means we write the expression as 3 * (45 + 15). Now, we just need to simplify the expression inside the parentheses. We add 45 and 15 together, which equals 60. So our expression becomes 3 * 60.

Finally, we multiply 3 by 60 to get our final answer, which is 180. Therefore, 3 * 45 + 3 * 15 = 180. Factoring out the 3 made it easier to simplify the expression and arrive at the solution. This method not only simplifies calculations but also provides a structured approach to problem-solving, which is essential in mathematics. Mastering such basic factorizations is key to handling more complex algebraic manipulations later on. This example showcases how identifying and extracting common factors can significantly streamline arithmetic operations.

Example D: 23 * 718 - 162 * 23

Now, let's tackle the next example: 23 * 718 - 162 * 23. Again, we need to identify the common factor. In this expression, the common factor is 23, as it appears in both 23 * 718 and 162 * 23.

To factor out the 23, we rewrite the expression as 23 * (718 - 162). This step involves using the distributive property in reverse, similar to the previous example. Next, we subtract 162 from 718. Doing this, we get 556. So now our expression is 23 * 556.

Finally, we multiply 23 by 556. When we do this multiplication, we get 12788. So, 23 * 718 - 162 * 23 = 12788. By factoring out the common factor, we simplified the calculation and found the solution more efficiently. This highlights the importance of recognizing common elements in expressions to streamline problem-solving in mathematics. Moreover, this technique provides a clear and organized method for tackling potentially complex calculations. The ability to identify and factor out common numbers is a fundamental skill that significantly aids in algebraic simplification.

Example G: 2413 * 1001 - 2413

Moving on to the next example: 2413 * 1001 - 2413. At first glance, it might seem like only one term has 2413, but remember that 2413 is the same as 2413 * 1. So, we can rewrite the expression as 2413 * 1001 - 2413 * 1. Now, it's clear that 2413 is the common factor.

We factor out the 2413, and rewrite the expression as 2413 * (1001 - 1). Next, we simplify the expression inside the parentheses. Subtracting 1 from 1001 gives us 1000. Thus, our expression becomes 2413 * 1000.

Now, we just need to multiply 2413 by 1000. This is simple: 2413 * 1000 = 2413000. Therefore, 2413 * 1001 - 2413 = 2413000. This example illustrates that sometimes you need to rewrite terms slightly to reveal the common factor. Recognizing such implicit common factors is a key skill in algebraic simplification and problem-solving. The process is straightforward once the common factor is identified, making the overall calculation much more manageable.

Example H: 20 * 48 + 20 * 38 + 20 * 15

Finally, let's look at the last example: 20 * 48 + 20 * 38 + 20 * 15. In this expression, the common factor is 20, as it appears in all three terms. To factor out the 20, we rewrite the expression as 20 * (48 + 38 + 15). Now, we need to simplify the expression inside the parentheses by adding 48, 38, and 15 together. 48 + 38 + 15 = 101, so our expression becomes 20 * 101.

To finish it off, we multiply 20 by 101, which equals 2020. Thus, 20 * 48 + 20 * 38 + 20 * 15 = 2020. Factoring out the common factor made the arithmetic much simpler and reduced the chances of making errors. This example also shows that the same principle applies even when there are more than two terms in the expression. By consistently identifying and factoring out common factors, we can greatly simplify mathematical problems. This ultimately helps in more efficient and accurate problem-solving.

Conclusion

Alright, guys, that wraps up our exploration of factoring with common factors! We've worked through several examples, each highlighting a slightly different aspect of this technique. Remember, the key is to always look for the common factor in each term and then use the distributive property in reverse to simplify the expression. Factoring is a fundamental skill in algebra, and mastering it will make your math life so much easier. Keep practicing, and you'll become a pro in no time! Keep up the great work, and happy factoring!