Simplifying Polynomial Expressions A² - (B + C) A Step-by-Step Guide

Polynomials are fundamental building blocks in algebra, and understanding how to manipulate them is crucial for success in mathematics. This article delves into a specific polynomial problem, providing a step-by-step solution and highlighting key concepts along the way. In this comprehensive exploration, we aim to demystify the process of polynomial operations, focusing specifically on solving the expression A² - (B + C) where A, B, and C are given polynomials. This problem not only reinforces fundamental algebraic principles but also serves as a practical exercise in applying these concepts. By breaking down each step and elucidating the underlying logic, we aim to empower readers with a solid understanding of polynomial manipulation. Whether you're a student grappling with algebra or simply seeking to refresh your knowledge, this article offers a detailed guide to mastering polynomial operations. Let's embark on this mathematical journey together, unraveling the intricacies of polynomial expressions and building a strong foundation for further algebraic explorations. We will begin by defining the given polynomials and then systematically work through the operations, ensuring clarity and precision in each step. This methodical approach will not only lead us to the solution but also enhance our understanding of how different polynomial operations interact with each other. Our goal is not just to arrive at the correct answer but to foster a deeper appreciation for the elegance and power of algebraic techniques.

Defining the Polynomials

Before we dive into the solution, let's clearly define the polynomials involved. We are given:

• A = 3x - 4
• B = x + 7
• C = x² + 2

These polynomials form the basis of our problem, and it's essential to have a firm grasp of their structure. Each polynomial is an expression consisting of variables (in this case, x) and constants, combined using addition, subtraction, and multiplication. The exponents of the variables are non-negative integers. Polynomial A is a linear polynomial, meaning the highest power of x is 1. Polynomial B is also a linear polynomial. Polynomial C is a quadratic polynomial, as the highest power of x is 2. Understanding the nature of these polynomials helps us anticipate the types of terms that will arise during our calculations. For instance, squaring polynomial A will likely introduce a quadratic term, while adding B and C will combine like terms. As we proceed, we'll see how these individual polynomials interact to form the final expression. The clarity in defining these polynomials is the first crucial step in solving the problem accurately and efficiently. By establishing a clear foundation, we minimize the risk of errors and enhance our understanding of the subsequent steps. This meticulous approach is characteristic of effective problem-solving in mathematics, where precision and attention to detail are paramount. So, let's keep these definitions in mind as we move forward, building upon this foundation to unravel the complexities of the expression A² - (B + C).

Step 1: Calculate A²

The first step in simplifying the expression is to calculate A². This means multiplying the polynomial A by itself:

A² = (3x - 4) * (3x - 4)

To perform this multiplication, we can use the distributive property (also known as the FOIL method for binomials): First, Outer, Inner, Last. This method ensures that each term in the first polynomial is multiplied by each term in the second polynomial. Applying the distributive property, we get:

A² = (3x * 3x) + (3x * -4) + (-4 * 3x) + (-4 * -4)

Now, let's simplify each term:

A² = 9x² - 12x - 12x + 16

Combine the like terms (-12x and -12x):

A² = 9x² - 24x + 16

So, we have found that A² is equal to 9x² - 24x + 16. This quadratic polynomial will be a crucial component in the next steps of our calculation. The process of squaring a binomial like (3x - 4) is a common operation in algebra, and mastering this technique is essential for simplifying more complex expressions. The distributive property is a powerful tool that allows us to systematically multiply polynomials, ensuring that no terms are missed. By breaking down the multiplication into smaller steps, we can minimize errors and gain a clearer understanding of the process. This careful and methodical approach is key to success in algebraic manipulations. Now that we have calculated A², we are one step closer to solving the original problem. We can move on to the next step, which involves adding polynomials B and C. Remember, each step builds upon the previous one, so accuracy and clarity are paramount. Let's continue our journey towards simplifying the expression A² - (B + C).

Step 2: Calculate B + C

Next, we need to calculate B + C. To add polynomials, we simply combine like terms. Recall that:

• B = x + 7
• C = x² + 2

Adding these polynomials, we get:

B + C = (x + 7) + (x² + 2)

Now, let's combine the like terms. We have a quadratic term (x²), a linear term (x), and constant terms (7 and 2). Combining the constants, we get:

B + C = x² + x + (7 + 2)

B + C = x² + x + 9

So, the sum of polynomials B and C is x² + x + 9. This result will be used in the final step when we subtract (B + C) from A². Adding polynomials involves identifying and combining terms with the same variable and exponent. This is a fundamental operation in algebra, and proficiency in this skill is essential for simplifying more complex expressions. The process is straightforward: we simply group the like terms and add their coefficients. In this case, we had a linear term (x) in polynomial B and a quadratic term (x²) in polynomial C. Since there were no corresponding terms in the other polynomial, they remained unchanged in the sum. The constant terms, however, could be combined to give a single constant term. This step-by-step approach ensures that we don't miss any terms and that we combine them correctly. Now that we have calculated both A² and (B + C), we are ready to perform the final subtraction. This will bring us to the simplest form of the expression A² - (B + C). Remember, each step we've taken has been a building block, leading us closer to the solution. Let's proceed with confidence and complete the final step.

Step 3: Calculate A² - (B + C)

Now we have A² and (B + C), we can subtract (B + C) from A². We found that:

• A² = 9x² - 24x + 16
• B + C = x² + x + 9

So, we need to calculate:

A² - (B + C) = (9x² - 24x + 16) - (x² + x + 9)

To subtract polynomials, we distribute the negative sign to each term in the second polynomial and then combine like terms:

A² - (B + C) = 9x² - 24x + 16 - x² - x - 9

Now, let's combine the like terms:

• Combine the x² terms: 9x² - x² = 8x²
• Combine the x terms: -24x - x = -25x
• Combine the constant terms: 16 - 9 = 7

Putting it all together, we get:

A² - (B + C) = 8x² - 25x + 7

Therefore, the simplest form of A² - (B + C) is 8x² - 25x + 7. This final step demonstrates the importance of careful distribution and combination of like terms when subtracting polynomials. The negative sign in front of the parentheses changes the sign of each term inside, so it's crucial to distribute it correctly. Once the distribution is done, the process of combining like terms is similar to addition, but with attention to the signs. We grouped the x² terms, the x terms, and the constant terms separately, and then performed the appropriate operations. This methodical approach ensures that we don't make any sign errors and that we combine the terms accurately. The result, 8x² - 25x + 7, is a quadratic polynomial in its simplest form. It cannot be factored further using simple methods, and there are no more like terms to combine. This completes our solution to the problem. We have successfully calculated A² - (B + C) by following a step-by-step approach, and we have arrived at the simplest form of the expression.

Final Answer

The simplest form of A² - (B + C) is:

8x² - 25x + 7

This is our final answer. We have successfully navigated through the polynomial operations, starting from defining the polynomials, calculating A², finding the sum of B and C, and finally, subtracting (B + C) from A². The result is a quadratic polynomial, 8x² - 25x + 7, which represents the simplified expression. Throughout this process, we have emphasized the importance of accuracy, clarity, and a step-by-step approach. Each step built upon the previous one, and by carefully performing each operation, we minimized the risk of errors. The use of the distributive property, the combination of like terms, and the proper handling of signs were all crucial elements in arriving at the correct answer. This exercise demonstrates the power of algebraic manipulation and the importance of mastering these fundamental skills. Polynomial operations are a cornerstone of algebra, and the ability to simplify expressions like A² - (B + C) is essential for success in more advanced topics. By understanding the underlying principles and practicing these techniques, we can build a strong foundation for future mathematical endeavors. The final answer, 8x² - 25x + 7, is not just a solution to a specific problem; it is a testament to the power of systematic problem-solving and the beauty of algebraic simplification. We hope this detailed explanation has provided valuable insights and has empowered you to tackle similar polynomial problems with confidence.

Conclusion

In conclusion, we have successfully simplified the expression A² - (B + C) to 8x² - 25x + 7. This process involved several key steps, including squaring a binomial, adding polynomials, and subtracting polynomials. Each step required careful attention to detail and a thorough understanding of algebraic principles. We began by clearly defining the polynomials A, B, and C, which formed the foundation of our problem. Then, we calculated A² using the distributive property, ensuring that each term was multiplied correctly. Next, we found the sum of B and C by combining like terms. Finally, we subtracted (B + C) from A², again paying close attention to the distribution of the negative sign and the combination of like terms. The final result, 8x² - 25x + 7, is a quadratic polynomial in its simplest form. This exercise has highlighted the importance of several fundamental algebraic skills, including the distributive property, combining like terms, and handling signs correctly. These skills are essential for simplifying more complex expressions and for solving a wide range of algebraic problems. By breaking down the problem into smaller, manageable steps, we were able to minimize errors and gain a clearer understanding of the process. This methodical approach is a valuable strategy for problem-solving in mathematics and in many other areas of life. We hope that this detailed explanation has provided you with a solid understanding of polynomial operations and has inspired you to explore further the fascinating world of algebra. Remember, practice is key to mastering these skills, so we encourage you to try similar problems and continue to build your mathematical confidence. The journey through algebra is a rewarding one, and each problem solved is a step forward in your mathematical development.