Factoring The Expression 5x(x+9)-7(x+9) A Step-by-Step Guide
Factoring expressions is a fundamental skill in algebra, allowing us to simplify complex equations and solve for unknown variables. In this article, we will delve into the process of factoring the expression $5x(x+9)-7(x+9)$ completely. We will break down each step, providing a clear and concise explanation to enhance your understanding. By the end of this guide, you'll be well-equipped to tackle similar factoring problems with confidence.
Understanding Factoring
Before we dive into the specifics of the given expression, let's briefly revisit the concept of factoring. Factoring is the process of breaking down an expression into its constituent factors, which are terms that multiply together to give the original expression. It's like reverse distribution, where we identify common elements and pull them out. Factoring simplifies expressions, making them easier to work with, and is crucial for solving equations, finding roots, and simplifying rational expressions. Think of factoring as the opposite of expanding; instead of multiplying terms out, we are looking for ways to group them back together.
The process of factoring often involves identifying common factors within the terms of an expression. These common factors can be numbers, variables, or even entire expressions. Once a common factor is identified, it can be factored out, leaving a simpler expression behind. This technique is particularly useful when dealing with polynomial expressions, as it allows us to reduce the degree of the polynomial and make it more manageable. Understanding factoring is not only essential for algebraic manipulations but also serves as a cornerstone for more advanced mathematical concepts.
Step-by-Step Factoring of 5x(x+9)-7(x+9)
1. Identifying the Common Factor
The first step in factoring the expression $5x(x+9)-7(x+9)$ is to identify the common factor. Upon close inspection, we can see that the term $(x+9)$ appears in both parts of the expression. This is our common factor. Identifying this common binomial factor is the key to simplifying the expression efficiently. Recognizing common factors is a crucial step in the factoring process, as it allows us to rewrite the expression in a more manageable form. In this case, noticing that $(x+9)$ is present in both terms opens the door to a straightforward factoring solution.
2. Factoring Out the Common Factor
Now that we've identified the common factor $(x+9)$, we can factor it out of the expression. This involves rewriting the expression by placing the common factor outside parentheses and including the remaining terms inside. When we factor out $(x+9)$, we are left with $5x$ from the first term and $-7$ from the second term. Therefore, factoring out $(x+9)$ gives us $(x+9)(5x-7)$. Factoring out the common term allows us to consolidate the expression and rewrite it in a product form, which is the hallmark of a factored expression. This step is essential in simplifying the expression and making it easier to analyze or solve.
3. Verifying the Factored Form
To ensure that we have factored the expression correctly, we can distribute the terms back to the original form. Multiplying $(x+9)$ by $(5x-7)$ should yield the original expression, $5x(x+9)-7(x+9)$. Let's perform this verification:
Now, let's expand the original expression to see if it matches:
Since both expressions match, we can be confident that our factored form is correct. This verification step is crucial in the factoring process, as it helps to ensure that we have not made any errors in our calculations. By expanding the factored form and comparing it to the original expression, we can confirm the accuracy of our work.
4. Final Factored Form
After factoring out the common factor and verifying our result, we arrive at the completely factored form of the expression. The expression $5x(x+9)-7(x+9)$ in completely factored form is $(5x-7)(x+9)$. This factored form represents the simplest way to express the original expression as a product of two binomials. The final factored form is concise, and it provides valuable insights into the structure of the expression. It allows for easier manipulation in subsequent algebraic operations, such as solving equations or simplifying rational expressions.
Common Mistakes to Avoid
When factoring expressions, it's easy to make mistakes if you're not careful. One common mistake is failing to identify the greatest common factor (GCF). Always look for the largest factor that divides all terms evenly. Another mistake is incorrectly distributing terms during verification. Double-check your multiplication to ensure accuracy. Additionally, some students may stop factoring prematurely, leaving a partially factored expression. Make sure to factor completely until no more common factors can be extracted. Avoiding these common mistakes will help you factor expressions accurately and efficiently.
Conclusion
In this article, we walked through the process of factoring the expression $5x(x+9)-7(x+9)$ completely. By identifying the common factor $(x+9)$, factoring it out, and verifying our result, we arrived at the factored form $(5x-7)(x+9)$. Factoring is a crucial skill in algebra, and mastering it will greatly enhance your ability to solve equations and simplify expressions. Remember to always look for common factors, factor them out carefully, and verify your answers. With practice, factoring will become second nature, allowing you to tackle more complex algebraic problems with ease.
By understanding the step-by-step process and avoiding common mistakes, you can confidently factor expressions and apply this skill to various mathematical contexts. Factoring is not just a mechanical process; it also enhances your problem-solving abilities and provides a deeper understanding of algebraic structures. So, continue practicing, explore different types of factoring problems, and watch your algebraic skills flourish. Happy factoring!
