Expanding Binomial Products A Step-by-Step Guide To (5r + 2)(3r - 4)

In the realm of mathematics, algebraic expressions form the bedrock of numerous concepts and applications. Mastering the manipulation of these expressions is crucial for success in higher-level mathematics and related fields. Among the fundamental operations is the expansion of binomial products, a technique that allows us to rewrite expressions in a more manageable form. In this article, we will delve into the expansion of the expression (5r + 2)(3r - 4), a classic example that showcases the application of the distributive property and the combination of like terms. Our journey will involve a step-by-step breakdown of the expansion process, highlighting the underlying principles and potential pitfalls to avoid. By the end of this exploration, you will not only be able to confidently expand similar expressions but also gain a deeper appreciation for the elegance and power of algebraic manipulation.

The cornerstone of expanding binomial products lies in the distributive property, a fundamental principle that governs how multiplication interacts with addition and subtraction. In its simplest form, the distributive property states that for any numbers a, b, and c, the following holds true: a(b + c) = ab + ac. This seemingly simple rule forms the basis for expanding more complex expressions. To understand the distributive property intuitively, consider it as a way of ensuring that each term within the parentheses is multiplied by the term outside. In the context of expanding binomial products, we extend this principle to encompass the multiplication of two binomials, each containing two terms. This process involves systematically multiplying each term in the first binomial by each term in the second binomial, ensuring that no term is left behind. The distributive property not only simplifies the expansion process but also provides a clear and logical framework for manipulating algebraic expressions. Grasping this principle is essential for mastering the expansion of binomial products and other related algebraic techniques. The key takeaway is that the distributive property allows us to break down complex multiplications into a series of simpler operations, ultimately leading to a more manageable expression.

Now, let's embark on the step-by-step expansion of the expression (5r + 2)(3r - 4). This process involves applying the distributive property systematically to ensure that each term in the first binomial is multiplied by each term in the second binomial.

Step 1: Distribute the first term

We begin by distributing the first term of the first binomial, which is 5r, across the terms of the second binomial: 5r * (3r - 4) = (5r * 3r) + (5r * -4) = 15r² - 20r. This step utilizes the distributive property to break down the multiplication into two simpler terms.

Step 2: Distribute the second term

Next, we distribute the second term of the first binomial, which is 2, across the terms of the second binomial: 2 * (3r - 4) = (2 * 3r) + (2 * -4) = 6r - 8. Again, we apply the distributive property to ensure that each term is properly multiplied.

Step 3: Combine the results

Now, we combine the results from the previous two steps: (15r² - 20r) + (6r - 8). This step involves bringing together the terms obtained from distributing both terms of the first binomial.

Step 4: Combine like terms

The final step involves identifying and combining like terms. Like terms are those that have the same variable raised to the same power. In this case, we have two like terms: -20r and 6r. Combining these terms, we get -20r + 6r = -14r. The other terms, 15r² and -8, do not have any like terms and remain unchanged. Therefore, the expanded expression becomes 15r² - 14r - 8. This final form represents the simplified version of the original expression, obtained through the systematic application of the distributive property and the combination of like terms.

Having meticulously expanded the expression (5r + 2)(3r - 4), we have arrived at the result 15r² - 14r - 8. Now, let's examine the multiple-choice options provided to identify the correct answer.

• A. 15r² + 8: This option is incorrect because it lacks the crucial middle term, -14r, and has the wrong sign for the constant term.
• B. 15r² - 8: This option is also incorrect as it omits the -14r term, which is essential for the correct expansion.
• C. 15r² + 14r - 8: This option has the correct terms but the wrong sign for the middle term. It shows +14r instead of -14r.
• D. 15r² - 14r - 8: This option perfectly matches our expanded result, containing all the correct terms with the accurate signs.

Therefore, the correct answer is D. 15r² - 14r - 8. This process of elimination not only confirms our solution but also reinforces the importance of careful calculation and attention to detail when expanding algebraic expressions. By systematically working through each step and comparing the result with the given options, we can confidently arrive at the correct answer.

Expanding binomial products can be a straightforward process, but certain common mistakes can lead to incorrect results. Being aware of these pitfalls can significantly improve accuracy and prevent errors.

• Forgetting to distribute: One of the most frequent errors is failing to distribute each term of the first binomial across all terms of the second binomial. This often results in missing terms in the expanded expression. For instance, incorrectly expanding (5r + 2)(3r - 4) might involve only multiplying 5r by 3r and 2 by -4, neglecting the cross-terms 5r * -4 and 2 * 3r. To avoid this, always ensure that each term in the first binomial is multiplied by each term in the second binomial.
• Sign errors: Sign errors are another common source of mistakes. When multiplying terms with negative signs, it's crucial to pay close attention to the rules of sign multiplication. A negative times a positive is negative, and a negative times a negative is positive. For example, in the expansion of (5r + 2)(3r - 4), the term 5r * -4 results in -20r, and failing to include the negative sign would lead to an incorrect answer. Double-checking signs at each step can help prevent these errors.
• Incorrectly combining like terms: After distributing, the next step is to combine like terms. A common mistake is to combine terms that are not like terms, such as adding 15r² to -14r. Like terms must have the same variable raised to the same power. Additionally, sign errors can occur when combining like terms, such as incorrectly adding -20r and 6r to get -26r instead of -14r. Carefully identifying and combining like terms, while paying attention to signs, is essential for obtaining the correct final expression.

By being mindful of these common mistakes and taking the necessary precautions, you can significantly improve your accuracy in expanding binomial products.

In this comprehensive exploration, we have delved into the expansion of the algebraic expression (5r + 2)(3r - 4), a fundamental exercise in algebraic manipulation. We began by understanding the distributive property, the cornerstone of expanding binomial products, and then systematically applied it in a step-by-step manner. We carefully distributed each term of the first binomial across the terms of the second binomial, combined the resulting terms, and finally, identified and combined like terms to arrive at the simplified expression 15r² - 14r - 8. This process not only demonstrates the mechanics of expansion but also highlights the importance of precision and attention to detail. We then compared our result with the provided multiple-choice options, confidently identifying option D as the correct answer. Furthermore, we addressed common mistakes to avoid, such as forgetting to distribute, sign errors, and incorrectly combining like terms, equipping you with the knowledge to navigate potential pitfalls. Mastering the expansion of binomial products is a crucial skill in algebra and serves as a building block for more advanced mathematical concepts. By understanding the underlying principles and practicing diligently, you can confidently tackle similar expressions and unlock the power of algebraic manipulation. The ability to expand binomial products not only enhances your problem-solving skills but also deepens your appreciation for the elegance and structure of mathematics. This skill is invaluable for various applications in mathematics, science, and engineering, making it a worthwhile investment of your time and effort.