Adding Polynomials Step-by-Step Solutions And Guide

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Polynomials are fundamental building blocks in algebra, and understanding how to manipulate them is crucial for success in mathematics. One of the most basic operations you'll perform with polynomials is addition. This article will serve as a comprehensive guide to adding polynomials, walking you through the process step-by-step with clear explanations and examples. We'll tackle five different polynomial addition problems, each designed to illustrate key concepts and techniques. By the end of this guide, you'll be able to confidently add polynomials of various complexities. In this comprehensive guide, we'll be focusing on polynomial addition, an essential operation in algebra. Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Adding polynomials involves combining like terms, which are terms that have the same variables raised to the same powers. Mastering this skill is crucial for simplifying expressions, solving equations, and understanding more advanced algebraic concepts. In the sections that follow, we will explore five different polynomial addition problems, each carefully chosen to illustrate various aspects of the process. We'll break down each step, providing clear explanations and helpful tips along the way. Whether you're a student just starting out with algebra or someone looking to refresh your skills, this guide will equip you with the knowledge and confidence to add polynomials effectively.

The first problem presents a straightforward example of adding two binomials (polynomials with two terms). To add these polynomials, we need to identify and combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have terms with the variable 'a' and terms with the variable 'b'. The given expression is (5a - 7b) + (2a - 5b). The initial step in adding these polynomials involves recognizing the like terms. Like terms are those that contain the same variables raised to the same powers. Here, '5a' and '2a' are like terms because they both contain the variable 'a' raised to the power of 1. Similarly, '-7b' and '-5b' are like terms because they both contain the variable 'b' raised to the power of 1. To combine like terms, we simply add or subtract their coefficients. The coefficient is the numerical part of the term. For the 'a' terms, we have 5a + 2a. Adding the coefficients 5 and 2 gives us 7, so the combined term is 7a. For the 'b' terms, we have -7b - 5b. Adding the coefficients -7 and -5 gives us -12, so the combined term is -12b. Now, we write the simplified expression by combining the results from the previous step. We have 7a from combining the 'a' terms and -12b from combining the 'b' terms. Putting these together, we get the simplified expression 7a - 12b. This is the sum of the two original polynomials. This problem illustrates the fundamental principle of adding polynomials: combining like terms. By carefully identifying and adding the coefficients of like terms, we can simplify complex expressions and arrive at the correct answer.

Step-by-step Solution

  1. Identify like terms: 5a and 2a are like terms. -7b and -5b are like terms.
  2. Combine like terms: (5a + 2a) + (-7b - 5b)
  3. Simplify: 7a - 12b

Therefore, (5a - 7b) + (2a - 5b) = 7a - 12b

This problem is similar to the first one, but it reinforces the importance of paying attention to the signs of the coefficients. We again have two binomials, and our goal is to add them by combining like terms. The expression we are working with is (-3a - 8b) + (4a - 4b). As with the previous problem, the first step is to identify the like terms. In this case, we have '-3a' and '4a' as like terms, both containing the variable 'a' raised to the power of 1. We also have '-8b' and '-4b' as like terms, both containing the variable 'b' raised to the power of 1. Remember that like terms must have the same variable and the same exponent. Once we've identified the like terms, the next step is to combine them. This involves adding or subtracting their coefficients. For the 'a' terms, we have -3a + 4a. Adding the coefficients -3 and 4 gives us 1, so the combined term is 1a, which is typically written as just 'a'. For the 'b' terms, we have -8b - 4b. Adding the coefficients -8 and -4 gives us -12, so the combined term is -12b. After combining the like terms, we write the simplified expression by putting the results together. We have 'a' from combining the 'a' terms and '-12b' from combining the 'b' terms. Putting these together, we get the simplified expression a - 12b. This is the sum of the two original polynomials. This problem further emphasizes the importance of careful attention to signs when adding polynomials. A mistake in adding or subtracting the coefficients can lead to an incorrect answer.

Step-by-step Solution

  1. Identify like terms: -3a and 4a are like terms. -8b and -4b are like terms.
  2. Combine like terms: (-3a + 4a) + (-8b - 4b)
  3. Simplify: a - 12b

Therefore, (-3a - 8b) + (4a - 4b) = a - 12b

This problem introduces terms with higher powers of variables, but the principle of adding like terms remains the same. Here, we have two binomials, and we need to find their sum. The given polynomials are (-10a²b² + 1) and (14a²b² + 8). The first step, as always, is to identify the like terms in the expression. In this case, we have '-10a²b²' and '14a²b²' as like terms. Both terms contain the variables 'a' and 'b', each raised to the power of 2. It's crucial that both the variables and their exponents match for terms to be considered like terms. We also have the constant terms '1' and '8' as like terms. Constant terms are those that do not contain any variables. Once we've identified the like terms, we proceed to combine them. For the 'a²b²' terms, we have -10a²b² + 14a²b². Adding the coefficients -10 and 14 gives us 4, so the combined term is 4a²b². Remember that when adding like terms, we only add or subtract the coefficients; the variables and their exponents remain the same. For the constant terms, we have 1 + 8. Adding these gives us 9. Constant terms can be directly added since they do not have any variable component. Now, we write the simplified expression by combining the results from the previous step. We have 4a²b² from combining the 'a²b²' terms and 9 from combining the constant terms. Putting these together, we get the simplified expression 4a²b² + 9. This is the sum of the two original polynomials. This problem highlights the importance of carefully matching variables and their exponents when identifying like terms.

Step-by-step Solution

  1. Identify like terms: -10a²b² and 14a²b² are like terms. 1 and 8 are like terms.
  2. Combine like terms: (-10a²b² + 14a²b²) + (1 + 8)
  3. Simplify: 4a²b² + 9

Therefore, (-10a²b² + 1) + (14a²b² + 8) = 4a²b² + 9

This problem involves polynomials with multiple terms and different combinations of variables. The core concept, however, remains the same: combine like terms. We are given the expression (-17x²y + 11xy - 2) + (-7x²y - 11xy - 8). The first step is to meticulously identify the like terms within the given expression. This requires careful attention to both the variables and their exponents. We have the following terms to consider: -17x²y, 11xy, -2, -7x²y, -11xy, and -8. Let's analyze each term: -17x²y and -7x²y are like terms because they both contain the variables 'x' squared and 'y' to the power of 1. 11xy and -11xy are like terms because they both contain the variables 'x' and 'y', each to the power of 1. -2 and -8 are like terms because they are both constant terms, meaning they don't contain any variables. Once we've correctly identified the like terms, we can proceed to combine them. For the 'x²y' terms, we have -17x²y - 7x²y. Adding the coefficients -17 and -7 gives us -24, so the combined term is -24x²y. For the 'xy' terms, we have 11xy - 11xy. Adding the coefficients 11 and -11 gives us 0, so the combined term is 0xy, which is simply 0. This means these terms effectively cancel each other out. For the constant terms, we have -2 - 8. Adding these gives us -10. Now that we've combined all the like terms, we can write the simplified expression. We have -24x²y from the 'x²y' terms, 0 from the 'xy' terms (which we don't need to write), and -10 from the constant terms. Putting these together, we get the simplified expression -24x²y - 10. This is the sum of the two original polynomials. This problem emphasizes the importance of being thorough and organized when identifying and combining like terms, especially when dealing with polynomials that have multiple variables and terms.

Step-by-step Solution

  1. Identify like terms: -17x²y and -7x²y are like terms. 11xy and -11xy are like terms. -2 and -8 are like terms.
  2. Combine like terms: (-17x²y - 7x²y) + (11xy - 11xy) + (-2 - 8)
  3. Simplify: -24x²y - 10

Therefore, (-17x²y + 11xy - 2) + (-7x²y - 11xy - 8) = -24x²y - 10

Our final problem involves adding two trinomials (polynomials with three terms). The approach remains consistent: identify and combine like terms. We are given the expression (m² - 3m - 5) + (m² + 10m - 8). The first crucial step is to identify the like terms within the expression. This involves carefully examining each term and grouping those that have the same variable raised to the same power. In this case, we have the following terms: m², -3m, -5, m², 10m, and -8. Let's break down the identification of like terms: m² and m² are like terms because they both contain the variable 'm' squared. -3m and 10m are like terms because they both contain the variable 'm' raised to the power of 1. -5 and -8 are like terms because they are both constant terms, meaning they do not contain any variables. With the like terms identified, we can proceed to combine them. For the 'm²' terms, we have m² + m². Adding the coefficients (which are both 1 in this case) gives us 2, so the combined term is 2m². For the 'm' terms, we have -3m + 10m. Adding the coefficients -3 and 10 gives us 7, so the combined term is 7m. For the constant terms, we have -5 - 8. Adding these gives us -13. Now that we've combined all the like terms, we can write the simplified expression. We have 2m² from the 'm²' terms, 7m from the 'm' terms, and -13 from the constant terms. Putting these together, we get the simplified expression 2m² + 7m - 13. This is the sum of the two original polynomials. This problem reinforces the importance of a systematic approach to identifying and combining like terms, ensuring that all terms are accounted for and correctly added.

Step-by-step Solution

  1. Identify like terms: m² and m² are like terms. -3m and 10m are like terms. -5 and -8 are like terms.
  2. Combine like terms: (m² + m²) + (-3m + 10m) + (-5 - 8)
  3. Simplify: 2m² + 7m - 13

Therefore, (m² - 3m - 5) + (m² + 10m - 8) = 2m² + 7m - 13

Adding polynomials is a fundamental skill in algebra. By consistently applying the principle of combining like terms, you can successfully add polynomials of any complexity. Remember to pay close attention to the signs of the coefficients and ensure that you are only combining terms with the same variables and exponents. With practice, you'll become proficient at this essential algebraic operation. Throughout this guide, we've worked through five different polynomial addition problems, each designed to illustrate key concepts and techniques. We've emphasized the importance of identifying like terms, combining their coefficients, and writing the simplified expression. By following the step-by-step solutions and understanding the underlying principles, you can confidently tackle a wide range of polynomial addition problems. Remember that consistent practice is key to mastering any mathematical skill. Work through additional examples, and don't hesitate to review the concepts and techniques presented in this guide as needed. With dedication and effort, you'll develop a strong foundation in polynomial addition, setting you up for success in more advanced algebraic topics. We hope this comprehensive guide has been helpful in enhancing your understanding of polynomial addition. Keep practicing, and you'll be well on your way to mastering this essential algebraic skill. Happy calculating!