Expression Represented By Factorization (3x+2)(3x-2)

This article will delve into the process of understanding which expression is represented by the factorization $(3x+2)(3x-2)$. This is a fundamental concept in algebra, often encountered when dealing with polynomials and their factored forms. We will explore the expansion of this expression, connecting it to a well-known algebraic identity, and ultimately determine the correct answer from the given options. This exploration will not only help you solve this specific problem but also strengthen your understanding of factorization and polynomial manipulation. By the end of this article, you should be able to confidently expand similar expressions and identify their equivalent forms.

Expanding the Factorized Expression

To determine the expression represented by the factorization $(3x+2)(3x-2)$, we need to expand it. This involves applying the distributive property, also known as the FOIL (First, Outer, Inner, Last) method. This method ensures that each term in the first binomial is multiplied by each term in the second binomial. Let's break down the steps:

1. First: Multiply the first terms of each binomial: $(3x) * (3x) = 9x^2$
2. Outer: Multiply the outer terms of the binomials: $(3x) * (-2) = -6x$
3. Inner: Multiply the inner terms of the binomials: $(2) * (3x) = 6x$
4. Last: Multiply the last terms of each binomial: $(2) * (-2) = -4$

Now, let's combine these results:

$9x^2 - 6x + 6x - 4$

Notice that the middle terms, $-6x$ and $+6x$, cancel each other out. This is a crucial observation, which leads us to simplify the expression further:

$9x^2 - 4$

This simplified expression, $9x^2 - 4$, represents the expanded form of the factorization $(3x+2)(3x-2)$. This process of expanding the expression highlights the relationship between factored forms and their polynomial equivalents. Understanding this relationship is vital for solving various algebraic problems, including simplifying expressions, solving equations, and identifying patterns.

Connecting to the Difference of Squares Identity

The expression we've just expanded, $9x^2 - 4$, should look familiar. It fits a specific pattern known as the difference of squares. This identity is a cornerstone of algebraic manipulation and is expressed as:

$(a + b)(a - b) = a^2 - b^2$

In our case, we can see that:

• $a = 3x$
• $b = 2$

If we substitute these values into the identity, we get:

$(3x + 2)(3x - 2) = (3x)^2 - (2)^2 = 9x^2 - 4$

This confirms our previous expansion and reinforces the application of the difference of squares identity. Recognizing this pattern allows for a quicker solution in similar problems. Instead of going through the full expansion using the distributive property, you can directly apply the identity to arrive at the answer. The difference of squares identity is a powerful tool in algebra, simplifying the factorization and expansion of expressions that fit this pattern. Mastering this identity will significantly enhance your problem-solving skills in algebra and related fields. Furthermore, the difference of squares identity is not just a mathematical trick; it represents a fundamental relationship between multiplication and subtraction, providing a concise way to express certain algebraic relationships. Understanding the underlying principles behind the identity will enable you to apply it more effectively in various mathematical contexts.

Analyzing the Answer Choices

Now that we've expanded the expression $(3x+2)(3x-2)$ and simplified it to $9x^2 - 4$, let's examine the given answer choices and identify the correct one:

• A. $9x^2 + 4$
• B. $9x^2 + 12x - 4$
• C. $9x^2 + 12x + 4$
• D. $9x^2 - 4$

By comparing our result, $9x^2 - 4$, with the answer choices, it's clear that option D is the correct answer. The other options contain terms that are not present in our simplified expression. For example, options B and C include the term $+12x$, which arises from an incorrect expansion or a misunderstanding of the difference of squares pattern. Option A has the correct terms but the wrong sign, adding instead of subtracting the constant term.

Common Mistakes to Avoid

When expanding and simplifying expressions like $(3x+2)(3x-2)$, it's crucial to be aware of common mistakes that can lead to incorrect answers. One frequent error is incorrectly applying the distributive property or the FOIL method. Forgetting to multiply each term in the first binomial by each term in the second can result in missing terms or incorrect coefficients. Another common mistake is misinterpreting the signs when multiplying. For instance, multiplying a positive term by a negative term should result in a negative term, but this can be easily overlooked. In the context of the difference of squares, a common error is to incorrectly assume that $(a+b)(a-b)$ is equal to $a^2 + b^2$ or to forget the subtraction sign altogether. It's also essential to pay close attention to the order of operations and simplify expressions correctly. Combining like terms is a critical step, and errors in this step can lead to an incorrect final answer. Avoiding these common pitfalls requires careful attention to detail and a solid understanding of algebraic principles. Practicing various examples and reviewing your work can help you identify and correct these mistakes, ultimately improving your accuracy and confidence in algebraic manipulations.

Conclusion

In conclusion, the expression represented by the factorization $(3x+2)(3x-2)$ is $9x^2 - 4$. We arrived at this answer by expanding the expression using the distributive property and simplifying the result. We also recognized the pattern of the difference of squares, which provided a more direct way to reach the solution. Understanding these algebraic principles and identities is essential for success in mathematics. This problem highlights the importance of careful expansion, simplification, and pattern recognition in algebra. By mastering these skills, you can confidently tackle similar problems and deepen your understanding of algebraic concepts. Remember to always double-check your work and be mindful of common mistakes. The key to success in algebra is consistent practice and a thorough understanding of the fundamental rules and identities. As you continue your mathematical journey, building a strong foundation in these core concepts will enable you to tackle more complex problems with ease and confidence. Furthermore, the skills you've developed in this process, such as logical reasoning and attention to detail, are transferable to many other areas of study and life, making the effort invested in mastering algebra truly worthwhile.

The correct answer is D. $9x^2 - 4$.