Expanding And Simplifying (3x² + 5n - 9)(3n - 5) A Step-by-Step Guide

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In this comprehensive guide, we will delve into the process of expanding and simplifying the algebraic expression (3x² + 5n - 9)(3n - 5). This type of problem is fundamental in algebra and requires a solid understanding of the distributive property and combining like terms. Whether you're a student learning algebra or someone looking to refresh your skills, this step-by-step breakdown will provide clarity and confidence in tackling similar problems.

Understanding the Basics

Before we dive into the expansion and simplification, let's briefly review the core concepts involved. The distributive property is the cornerstone of expanding expressions, and it states that a(b + c) = ab + ac. This property allows us to multiply a term by a sum or difference by distributing the term across each element within the parentheses. Combining like terms is another crucial step, where we group terms with the same variable and exponent and add or subtract their coefficients. For example, 3x² and 5x² are like terms and can be combined to 8x², while 3x² and 5x are not like terms and cannot be combined directly.

In our expression (3x² + 5n - 9)(3n - 5), we have two sets of parentheses, each containing multiple terms. To expand this, we will apply the distributive property multiple times, ensuring that each term in the first set of parentheses is multiplied by each term in the second set. This process is often referred to as the FOIL method (First, Outer, Inner, Last) when dealing with two binomials (expressions with two terms), but it's essentially an application of the distributive property. However, in our case, we have a trinomial (3x² + 5n - 9) multiplied by a binomial (3n - 5), so we'll need to be systematic in our distribution to ensure we don't miss any terms. The ultimate goal is to simplify the expression by combining like terms after the expansion. By meticulously following each step, we'll transform the initial expression into its most concise form, making it easier to analyze and work with in further mathematical operations or applications. This meticulous approach not only simplifies the expression but also enhances our understanding of algebraic manipulations and problem-solving strategies.

Step 1: Expanding the Expression

The first step in simplifying the expression (3x² + 5n - 9)(3n - 5) is to expand it using the distributive property. This involves multiplying each term in the first parentheses (3x² + 5n - 9) by each term in the second parentheses (3n - 5). Let's break it down systematically:

  • Multiply 3x² by (3n - 5): This gives us (3x² * 3n) - (3x² * 5) = 9x²n - 15x².
  • Multiply 5n by (3n - 5): This gives us (5n * 3n) - (5n * 5) = 15n² - 25n.
  • Multiply -9 by (3n - 5): This gives us (-9 * 3n) - (-9 * 5) = -27n + 45.

Now, we combine all these results to get the expanded expression: 9x²n - 15x² + 15n² - 25n - 27n + 45. This expanded form represents the initial multiplication, but it's not yet in its simplest form. We still have terms that might be like terms, and combining them is the next crucial step in our simplification process. The expanded expression reveals all the individual products resulting from the distribution, setting the stage for the subsequent step where we identify and combine like terms. It's essential to double-check this expanded form to ensure accuracy, as any errors here will propagate through the rest of the simplification. The goal is to transform the complex product into a sum of individual terms, each representing a different combination of the original variables and constants. By carefully expanding the expression, we pave the way for a more streamlined and manageable form that is easier to analyze and manipulate.

Step 2: Identifying Like Terms

After expanding the expression, we have 9x²n - 15x² + 15n² - 25n - 27n + 45. The next crucial step is to identify like terms. Like terms are terms that have the same variable(s) raised to the same power. In our expanded expression, we can see that:

  • -25n and -27n are like terms because they both have the variable 'n' raised to the power of 1.
  • The other terms (9x²n, -15x², 15n², and 45) do not have any like terms in the expression.

Identifying like terms is a fundamental skill in algebra, as it allows us to combine and simplify expressions effectively. It involves carefully examining the variables and their exponents in each term. The process of identifying like terms is akin to sorting objects into categories based on their properties. In this case, the 'properties' are the variables and their exponents. Once we have correctly identified the like terms, we can proceed to combine them, reducing the complexity of the expression and making it easier to work with. This step is essential for achieving the simplest form of the expression, which is crucial for further mathematical operations or applications. By focusing on the common variable and exponent patterns, we ensure that we group the appropriate terms together, paving the way for accurate and efficient simplification.

Step 3: Combining Like Terms

Now that we've identified the like terms in the expression 9x²n - 15x² + 15n² - 25n - 27n + 45, we can proceed to combine them. This involves adding or subtracting the coefficients of the like terms while keeping the variable and exponent the same. In our case, we only have one pair of like terms: -25n and -27n.

Combining -25n and -27n, we get -25n - 27n = -52n. This is because when we add two negative numbers, we add their absolute values and keep the negative sign.

Now, we rewrite the expression with the combined like terms: 9x²n - 15x² + 15n² - 52n + 45. This expression is now simplified as much as possible because there are no more like terms to combine. The process of combining like terms is a core principle in algebra, allowing us to reduce the complexity of expressions and make them easier to understand and manipulate. By adding or subtracting the coefficients of like terms, we effectively consolidate multiple terms into a single term, simplifying the overall expression. This step is crucial for arriving at the most concise form of the expression, which is essential for solving equations, evaluating expressions, and other algebraic operations. The result, 9x²n - 15x² + 15n² - 52n + 45, represents the simplified form of the original expression, achieved through careful expansion and combination of like terms.

Final Simplified Expression

After expanding and combining like terms, the simplified form of the expression (3x² + 5n - 9)(3n - 5) is 9x²n - 15x² + 15n² - 52n + 45. This expression is now in its simplest form, as there are no more like terms to combine, and it represents the most concise way to express the original product. This final form is crucial for various mathematical applications, such as solving equations, graphing functions, or further algebraic manipulations. The process of expanding and simplifying algebraic expressions is a fundamental skill in mathematics, enabling us to transform complex expressions into more manageable forms. The final simplified expression, 9x²n - 15x² + 15n² - 52n + 45, is a testament to the power of algebraic techniques in reducing complexity and revealing the underlying structure of mathematical relationships. By systematically applying the distributive property and combining like terms, we have successfully transformed the initial product into a simplified polynomial, ready for further analysis or application. This skill is not only essential for academic pursuits but also for practical problem-solving in various fields that rely on mathematical modeling and analysis.

Conclusion

In conclusion, expanding and simplifying the algebraic expression (3x² + 5n - 9)(3n - 5) involves a systematic application of the distributive property and the combination of like terms. We began by expanding the expression, multiplying each term in the first parentheses by each term in the second parentheses. This resulted in a longer expression with multiple terms. Next, we identified like terms, which are terms with the same variable(s) raised to the same power. Finally, we combined these like terms by adding or subtracting their coefficients. The final simplified expression, 9x²n - 15x² + 15n² - 52n + 45, represents the most concise form of the original expression. Mastering this process is essential for success in algebra and further mathematical studies. It provides a foundation for solving more complex equations and understanding mathematical relationships. The ability to manipulate algebraic expressions efficiently and accurately is a valuable skill that extends beyond the classroom, finding applications in various fields, including science, engineering, and economics. By understanding the underlying principles and practicing regularly, anyone can become proficient in expanding and simplifying algebraic expressions, unlocking a powerful tool for mathematical problem-solving and analysis.