What Is The Result Of Multiplying The Binomials (6r - 1) And (-8r - 3)?

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In the realm of algebra, polynomial multiplication is a fundamental operation. This article delves into the process of finding the product of two binomials, (6r−1)(−8r−3){ (6r - 1)(-8r - 3) }, and provides a step-by-step explanation to enhance your understanding. This exploration is crucial for anyone studying algebra, as it reinforces the distributive property and the combination of like terms. Mastering this skill is essential for solving more complex algebraic problems and is a cornerstone of mathematical proficiency. Understanding polynomial multiplication not only helps in academic settings but also has practical applications in various fields, such as engineering, economics, and computer science. The ability to manipulate and simplify algebraic expressions is a valuable asset in problem-solving and critical thinking. This article aims to break down the process in a clear and concise manner, making it accessible to learners of all levels.

Understanding the Problem

At the heart of our problem lies the task of multiplying two binomials: (6r−1){ (6r - 1) } and (−8r−3){ (-8r - 3) }. A binomial is a polynomial with two terms, and multiplying binomials is a common operation in algebra. The key to solving this problem is the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that each term in the first binomial is multiplied by each term in the second binomial. Before we dive into the solution, it's important to understand the structure of the problem. We have two expressions, each containing a variable term and a constant term. The multiplication process involves combining these terms in a specific way to arrive at the final product. This process is not just about applying a formula; it's about understanding the underlying principles of algebraic manipulation. By mastering this, you'll be better equipped to tackle more complex problems involving polynomial multiplication and simplification. The distributive property is a fundamental concept, and its application here serves as a building block for more advanced algebraic techniques. This section sets the stage for a detailed exploration of the solution, ensuring that the reader is well-prepared to follow the steps and understand the reasoning behind them.

Step-by-Step Solution

To find the product of (6r−1)(−8r−3){ (6r - 1)(-8r - 3) }, we will use the FOIL method, which stands for First, Outer, Inner, Last. This method is a systematic way to apply the distributive property to binomial multiplication. It ensures that every term in the first binomial is multiplied by every term in the second binomial. Let's break down the process step by step:

  1. First: Multiply the first terms of each binomial: (6r)×(−8r)=−48r2{ (6r) \times (-8r) = -48r^2 } This step involves multiplying the terms that appear first in each binomial. It's a straightforward application of the rules of multiplication and exponentiation. The result, −48r2{ -48r^2 }, is the first term of our final product. This step is crucial as it sets the foundation for the rest of the calculation. A clear understanding of how this term is derived is essential for grasping the overall process.

  2. Outer: Multiply the outer terms of the binomials: (6r)×(−3)=−18r{ (6r) \times (-3) = -18r } Here, we multiply the terms that are farthest apart in the expression. This step is another direct application of multiplication, and the result, −18r{ -18r }, is the second term of our product. This step is as important as the first, as it ensures that all necessary combinations of terms are accounted for. The ability to correctly identify and multiply these outer terms is key to accurate binomial multiplication.

  3. Inner: Multiply the inner terms of the binomials: (−1)×(−8r)=8r{ (-1) \times (-8r) = 8r } In this step, we multiply the terms that are closest to each other in the expression. The result, 8r{ 8r }, is the third term of our product. This step, along with the previous two, completes the multiplication of individual terms. Understanding the concept of inner terms and their multiplication is vital for mastering the FOIL method.

  4. Last: Multiply the last terms of each binomial: (−1)×(−3)=3{ (-1) \times (-3) = 3 } Finally, we multiply the last terms of each binomial. This step yields the constant term, 3, which is the final term of our product. This step ensures that all possible term combinations have been considered, completing the multiplication process.

Now, we add all the products together:

−48r2−18r+8r+3{ -48r^2 - 18r + 8r + 3 }

Combine like terms (the terms with the same variable and exponent):

−48r2+(−18r+8r)+3{ -48r^2 + (-18r + 8r) + 3 }

−48r2−10r+3{ -48r^2 - 10r + 3 }

Thus, the product of (6r−1)(−8r−3){ (6r - 1)(-8r - 3) } is −48r2−10r+3{ -48r^2 - 10r + 3 }. This final step is crucial as it simplifies the expression to its most basic form. Combining like terms is a fundamental algebraic skill, and its application here demonstrates the importance of this step in the overall solution.

Identifying the Correct Answer

After performing the multiplication and simplification, we have arrived at the expression −48r2−10r+3{ -48r^2 - 10r + 3 }. Now, we need to compare this result with the given options to identify the correct answer. This step is crucial to ensure that our calculations are accurate and that we select the appropriate option. Let's examine the options provided:

  • A. −48r2−10r+3{ -48r^2 - 10r + 3 }
  • B. −48r2−10r−3{ -48r^2 - 10r - 3 }
  • C. −48r2+3{ -48r^2 + 3 }
  • D. −48r2−3{ -48r^2 - 3 }

By comparing our result with the options, it is clear that option A, −48r2−10r+3{ -48r^2 - 10r + 3 }, matches our calculated product. Therefore, option A is the correct answer. This step highlights the importance of careful comparison and attention to detail. It's not enough to simply perform the calculations; we must also verify our result against the given options to ensure accuracy. This process reinforces the importance of double-checking our work and confirming that our solution aligns with the available choices.

Common Mistakes to Avoid

When multiplying binomials, several common mistakes can occur. Being aware of these potential pitfalls can help you avoid errors and ensure accurate results. This section aims to highlight these common mistakes and provide tips on how to avoid them. One frequent mistake is incorrectly applying the distributive property. For example, students might forget to multiply all terms in the first binomial by all terms in the second binomial. This often leads to missing terms in the final product. To avoid this, always use the FOIL method systematically, ensuring that each term is accounted for.

Another common error is making mistakes with signs. For instance, multiplying two negative terms should result in a positive term, but this can sometimes be overlooked. To prevent sign errors, pay close attention to the signs of each term and double-check your work. A simple way to do this is to write out each step clearly and carefully, paying attention to the positive and negative signs.

Combining like terms incorrectly is another frequent mistake. Remember that like terms must have the same variable and exponent. For example, −18r{ -18r } and 8r{ 8r } are like terms and can be combined, but −48r2{ -48r^2 } cannot be combined with them. To avoid this error, carefully identify like terms and combine them accurately.

Finally, a lack of attention to detail can lead to errors. This includes simple arithmetic mistakes or miscopying terms. To minimize these errors, take your time, work neatly, and double-check each step of your work. By being mindful of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in multiplying binomials. This section serves as a valuable guide for learners, helping them to identify potential pitfalls and develop strategies for avoiding them.

Practice Problems

To solidify your understanding of multiplying binomials, working through practice problems is essential. This section provides additional problems that you can use to test your skills and reinforce the concepts discussed in this article. Practice is key to mastering any mathematical skill, and binomial multiplication is no exception. The more you practice, the more comfortable and confident you will become with the process.

Here are a few practice problems:

  1. Find the product of (2x+3)(x−4){ (2x + 3)(x - 4) }.
  2. Multiply (5y−2)(3y+1){ (5y - 2)(3y + 1) }.
  3. Determine the product of (4a+5)(2a−3){ (4a + 5)(2a - 3) }.
  4. What is the result of (7b−1)(b+2){ (7b - 1)(b + 2) }?

For each problem, follow the steps outlined in the solution section. Use the FOIL method to multiply the binomials, and then combine like terms to simplify the expression. After solving each problem, check your answers to ensure accuracy. If you encounter any difficulties, review the step-by-step solution provided earlier in this article. Remember, the goal is not just to get the correct answer, but also to understand the process and reasoning behind it.

Working through these practice problems will not only enhance your skills in binomial multiplication but also improve your overall algebraic proficiency. Practice helps to build a deeper understanding of the concepts and allows you to apply them more effectively in different contexts. This section provides a valuable opportunity for learners to test their knowledge and reinforce their understanding through hands-on experience.

Conclusion

In conclusion, finding the product of (6r−1)(−8r−3){ (6r - 1)(-8r - 3) } involves applying the distributive property, commonly known as the FOIL method. The correct answer is −48r2−10r+3{ -48r^2 - 10r + 3 }, which corresponds to option A. This article has provided a detailed step-by-step solution, highlighting the importance of each step in the process. Understanding binomial multiplication is a fundamental skill in algebra, and mastering it can open doors to more advanced mathematical concepts. Throughout this article, we have emphasized the importance of the distributive property, the FOIL method, and the combination of like terms. These are essential tools for solving not just binomial multiplication problems, but a wide range of algebraic equations and expressions. By following the steps outlined and avoiding common mistakes, you can confidently tackle similar problems and achieve accurate results.

Furthermore, we have stressed the significance of practice in solidifying your understanding. The practice problems provided offer an opportunity to apply the concepts learned and reinforce your skills. Consistent practice is key to building fluency and confidence in mathematics. Finally, remember that algebra is a building block for many other areas of mathematics and science. The skills you develop in algebra will serve you well in future studies and in various real-world applications. This article aims to provide a comprehensive guide to binomial multiplication, equipping learners with the knowledge and skills they need to succeed. By mastering this fundamental concept, you can build a strong foundation for further exploration in mathematics.