Factoring: Rewrite 5x + 25 With Common Factors
Hey guys! Let's dive into factoring the expression $5x + 25$ using the greatest common factor. Factoring is a fundamental concept in algebra, and mastering it will help you simplify expressions and solve equations more efficiently. In this article, we'll break down the process step-by-step, making it super easy to understand. So, grab your pencils, and let's get started!
Understanding Factoring
Before we jump into the specific problem, let's quickly recap what factoring is all about. Factoring is the process of breaking down an expression into a product of its factors. Think of it as the reverse of expanding or distributing. For example, if we have $2(x + 3)$, expanding it gives us $2x + 6$. Factoring, on the other hand, would take $2x + 6$ back to $2(x + 3)$. The goal is to identify common elements in each term of the expression and pull them out.
The greatest common factor (GCF) is the largest number or expression that divides evenly into all terms. Finding the GCF is the key to factoring efficiently. Once you identify the GCF, you can rewrite the original expression as a product of the GCF and the remaining factors.
Now, why is factoring so important? Well, it simplifies algebraic expressions, which makes them easier to work with. It's also crucial for solving equations, especially quadratic equations, where factoring can help you find the roots. Plus, factoring pops up in various areas of math and science, so getting a solid grasp on it is super beneficial.
Let's not forget about real-world applications. Factoring can be used in optimization problems, such as maximizing area or minimizing costs. It's also useful in engineering, physics, and computer science. For instance, engineers might use factoring to simplify equations that describe the behavior of circuits or structures. In computer science, factoring can be applied in cryptography and data compression.
So, whether you're a student tackling algebra or a professional dealing with complex equations, mastering factoring is a skill that will serve you well. Let's move on to our specific problem and see how to apply these concepts.
Identifying the Common Factor in $5x + 25$
Okay, let's tackle the expression $5x + 25$. Our mission is to find the greatest common factor (GCF) of the terms $5x$ and $25$. When you look at these terms, what number divides both of them evenly? If you said 5, you're spot on!
The term $5x$ has factors of 5 and $x$, while the term $25$ has factors of 1, 5, and 25. The only common factor they share is 5. So, the GCF of $5x$ and $25$ is 5. Now that we've identified the GCF, we can rewrite the expression by factoring out this common factor. This involves dividing each term by the GCF and placing the GCF outside the parentheses.
Factoring out the 5 from $5x$ gives us $x$, because $5x \div 5 = x$. Similarly, factoring out the 5 from $25$ gives us $5$, because $25 \div 5 = 5$. Now, we write the factored expression as the GCF multiplied by the sum of the remaining terms inside the parentheses.
So, $5x + 25$ becomes $5(x + 5)$. This means we've successfully rewritten the expression using the common factor of 5. To check our work, we can distribute the 5 back into the parentheses: $5(x + 5) = 5x + 25$. Since this matches the original expression, we know we've factored it correctly.
Factoring out the GCF is a crucial skill in algebra. It simplifies expressions and makes them easier to work with. By identifying the common factors in each term, you can rewrite the expression in a more manageable form. This is particularly useful when solving equations or simplifying complex algebraic expressions.
Remember, the key is to always look for the largest number or expression that divides evenly into all terms. This ensures you're factoring out the greatest common factor. Practice makes perfect, so the more you factor, the better you'll become at it.
Rewriting the Expression
Now that we've identified the common factor, let's rewrite the expression $5x + 25$. We know that the greatest common factor (GCF) is 5. To rewrite the expression, we'll factor out the 5 from both terms.
First, divide $5x$ by 5, which gives us $x$. Then, divide $25$ by 5, which gives us 5. Now, place the GCF (5) outside the parentheses and write the remaining terms inside the parentheses. This gives us $5(x + 5)$.
So, the rewritten expression is $5(x + 5)$. This means that $5x + 25$ is equivalent to $5(x + 5)$. To verify, we can distribute the 5 back into the parentheses: $5 * x + 5 * 5 = 5x + 25$. Since this matches the original expression, we know we've done it correctly.
Factoring out the GCF is a valuable skill in algebra. It simplifies expressions and makes them easier to work with. By identifying the common factors in each term, you can rewrite the expression in a more manageable form. This is especially useful when solving equations or simplifying complex algebraic expressions.
It's important to note that the GCF must be the largest number or expression that divides evenly into all terms. This ensures you're factoring out the greatest common factor. Practice makes perfect, so the more you factor, the better you'll become at it.
In summary, to rewrite $5x + 25$ using a common factor, we identify the GCF (5), divide each term by the GCF, and write the expression as the GCF multiplied by the sum of the remaining terms inside the parentheses. This gives us $5(x + 5)$, which is the factored form of the original expression.
Selecting the Correct Option
Alright, let's circle back to the original question. We were asked to rewrite $5x + 25$ using a common factor, and we were given four options to choose from:
A. $5(x+5)$ B. $5(x+25)$ C. $5x(x+5)$ D. $5x(x+25)$
We've already determined that the correct way to rewrite $5x + 25$ using a common factor is $5(x + 5)$. Let's quickly analyze why the other options are incorrect.
Option B, $5(x + 25)$, is incorrect because when you distribute the 5, you get $5x + 125$, which is not equal to the original expression $5x + 25$.
Option C, $5x(x + 5)$, is incorrect because when you distribute the $5x$, you get $5x^2 + 25x$, which is also not equal to the original expression $5x + 25$.
Option D, $5x(x + 25)$, is incorrect because when you distribute the $5x$, you get $5x^2 + 125x$, which is, again, not equal to the original expression $5x + 25$.
Therefore, the correct answer is A. $5(x + 5)$. This is the only option that, when distributed, gives us the original expression $5x + 25$. Factoring is all about finding the common elements and rewriting the expression in a simplified form. By practicing these steps, you'll become more confident in your ability to factor expressions accurately.
So, there you have it! Factoring $5x + 25$ using a common factor is straightforward once you identify the greatest common factor and rewrite the expression accordingly. Always double-check your work by distributing the factored expression to ensure it matches the original. Keep practicing, and you'll become a factoring pro in no time!
Conclusion
In conclusion, factoring the expression $5x + 25$ involves identifying the greatest common factor (GCF), which in this case is 5. We then rewrite the expression by factoring out the 5 from both terms, resulting in $5(x + 5)$. This process simplifies the expression and makes it easier to work with in various algebraic manipulations.
We discussed the importance of factoring and its applications in simplifying algebraic expressions, solving equations, and various real-world scenarios. Understanding factoring is a fundamental skill in algebra that can be applied in numerous fields, from engineering to computer science.
By identifying the common factors and rewriting the expression, we ensure that the factored form is equivalent to the original expression. Always remember to double-check your work by distributing the factored expression to verify that it matches the original.
So, keep practicing factoring, and you'll become more proficient in simplifying and solving algebraic problems. Mastering this skill will undoubtedly enhance your understanding and capabilities in mathematics and related fields.