Factoring Expressions How To Factor 5x(x+9)-7(x+9)

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In the realm of mathematics, factoring expressions stands out as a fundamental skill. It not only simplifies complex equations but also provides a deeper understanding of the underlying relationships between variables and constants. In this article, we will delve into the intricacies of factoring, focusing on a specific expression: 5x(x+9)-7(x+9). Our journey will be methodical, breaking down each step to ensure clarity and comprehension. Whether you're a student grappling with algebra or simply someone looking to refresh their mathematical prowess, this guide is tailored to meet your needs.

Understanding the Basics of Factoring

Before we tackle the main expression, let's lay a solid foundation by revisiting the basics of factoring. Factoring, at its core, is the reverse process of expanding. When we expand, we multiply terms together, often using the distributive property. Factoring, on the other hand, involves identifying common factors within an expression and extracting them to rewrite the expression as a product of simpler terms. This skill is invaluable in solving equations, simplifying algebraic fractions, and tackling more advanced mathematical concepts. Understanding the fundamental principles of factoring is crucial. It allows you to break down complex expressions into simpler, more manageable forms. This not only makes problem-solving easier but also enhances your understanding of mathematical relationships. In essence, factoring is like reverse engineering an expression, uncovering its building blocks.

Common Factoring Techniques

There are several techniques used in factoring, each suited to different types of expressions. Some of the most common include:

  • Greatest Common Factor (GCF): Identifying the largest factor that divides all terms in the expression.
  • Difference of Squares: Factoring expressions in the form a² - b² into (a + b)(a - b).
  • Perfect Square Trinomials: Recognizing and factoring expressions like a² + 2ab + b² or a² - 2ab + b².
  • Factoring by Grouping: A technique used for expressions with four or more terms, where terms are grouped to reveal common factors.

Mastering these techniques is essential for effectively factoring a wide range of expressions. Each technique offers a unique approach to simplifying expressions, and knowing when to apply each one is a key skill in algebra. By understanding the underlying principles and practicing these methods, you can confidently tackle even the most challenging factoring problems.

Step-by-Step Factoring of 5x(x+9)-7(x+9)

Now, let's apply our factoring knowledge to the expression 5x(x+9)-7(x+9). This expression presents a classic scenario for factoring, and by working through it step by step, we can illustrate the key principles in action. This process not only simplifies the expression but also demonstrates the power of factoring in revealing the underlying structure of mathematical expressions. By carefully analyzing each step, you will gain a deeper understanding of how to approach similar factoring problems in the future.

Step 1: Identifying the Common Factor

The first step in factoring 5x(x+9)-7(x+9) is to identify the common factor. Looking at the expression, we can see that both terms, 5x(x+9) and -7(x+9), share a common factor of (x+9). This is a crucial observation, as it allows us to rewrite the expression in a more factored form. Identifying common factors is a fundamental skill in factoring, and it often serves as the starting point for simplifying complex expressions. Recognizing patterns and shared elements is key to unlocking the factored form of an expression. In this case, the presence of (x+9) in both terms is the key to our next step.

Step 2: Factoring out the Common Factor

Having identified (x+9) as the common factor, we can now factor it out of the expression. This involves rewriting the expression as a product of (x+9) and the remaining terms. When we factor out (x+9) from 5x(x+9), we are left with 5x. Similarly, when we factor out (x+9) from -7(x+9), we are left with -7. This process effectively isolates the common factor, allowing us to simplify the expression into a more manageable form. Factoring out the common factor is a core technique in algebra, and it often leads to significant simplification of expressions.

Step 3: Writing the Factored Expression

After factoring out the common factor, we can now write the expression in its factored form. This involves combining the remaining terms within parentheses. In our case, after factoring out (x+9), we are left with 5x and -7. Combining these terms, we get (5x - 7). Therefore, the factored expression becomes (5x - 7)(x + 9). This is the complete factored form of the original expression, and it represents a significant simplification. Writing the factored expression is the culmination of the factoring process, and it provides a clear and concise representation of the original expression.

Solution and Answer Options

Therefore, the completely factored form of the expression 5x(x+9)-7(x+9) is (5x-7)(x+9). Looking at the answer options provided:

  • A. (x+9)(x+9)
  • B. (5x-7)(x+9)
  • C. (5x-7)(x+9)(x+9)
  • D. (5x^2+45x)(-7x-63)

The correct answer is B. (5x-7)(x+9). This option matches the factored form we derived through our step-by-step process. Identifying the correct answer among the options requires a careful comparison between the factored form and the given choices. In this case, option B clearly aligns with our result, confirming the accuracy of our factoring process.

Common Mistakes to Avoid

Factoring can be tricky, and it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  • Incorrectly identifying the common factor: Make sure you're extracting the greatest common factor, not just any common factor.
  • Forgetting to include all terms: When factoring out a common factor, ensure that you account for all terms in the expression.
  • Errors in sign: Pay close attention to signs, especially when dealing with negative terms.
  • Over-factoring: Avoid factoring beyond the simplest form. Once you've factored out the GCF, check if further factoring is possible.
  • Not checking your work: Always double-check your factored expression by expanding it to see if it matches the original expression.

Avoiding these common mistakes is crucial for accurate factoring. By being mindful of these potential pitfalls, you can improve your factoring skills and reduce the likelihood of errors. Practice and attention to detail are key to mastering factoring and avoiding these common mistakes.

Conclusion

Factoring expressions is a vital skill in algebra and beyond. By understanding the basic techniques and practicing regularly, you can master this skill and confidently tackle a wide range of mathematical problems. In this article, we've walked through the process of factoring the expression 5x(x+9)-7(x+9), highlighting the key steps and common mistakes to avoid. Remember, the key to success in factoring, as in any mathematical endeavor, is practice, patience, and a willingness to learn from your mistakes. Factoring is not just a mechanical process; it's a way of seeing the structure and relationships within mathematical expressions. By mastering factoring, you are not just solving problems; you are gaining a deeper understanding of the language of mathematics.

By consistently applying these principles and techniques, you'll not only be able to factor expressions with ease but also develop a stronger foundation for more advanced mathematical concepts. Keep practicing, and you'll find that factoring becomes second nature, a powerful tool in your mathematical arsenal.