Finding The Sum Of Polynomials A Step By Step Guide

JU07/08/2025 08, 2025 by THE IDEN 52 views

Find the Sum: A Comprehensive Guide to Polynomial Addition

In the realm of mathematics, polynomial addition is a fundamental operation. This guide will provide a comprehensive exploration of how to find the sum of polynomials, focusing on the specific example: (2a² + ab + 2b) + (4a² - 3ab + 9). Understanding the process of combining like terms and simplifying expressions is crucial for success in algebra and beyond.

Polynomial addition is a fundamental operation in algebra that involves combining like terms from two or more polynomials. To successfully add polynomials, one must understand the concept of like terms, which are terms that have the same variables raised to the same powers. This guide aims to provide a detailed, step-by-step explanation of how to add polynomials, focusing on the given expression: (2a² + ab + 2b) + (4a² - 3ab + 9). By mastering this process, students and math enthusiasts can build a strong foundation for more advanced algebraic concepts.

Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Examples of polynomials include 2x² + 3x - 1, 4y³ - 2y + 5, and the expression we will focus on in this guide: (2a² + ab + 2b) + (4a² - 3ab + 9). Each term in a polynomial consists of a coefficient (a number) and a variable raised to a power. For instance, in the term 2a², 2 is the coefficient, and a² is the variable part. When adding polynomials, the key is to identify and combine like terms. Like terms are those that have the same variable(s) raised to the same power(s). For example, 2a² and 4a² are like terms because they both have the variable 'a' raised to the power of 2. Similarly, ab and -3ab are like terms because they both have the variables 'a' and 'b' raised to the power of 1. Terms like 2b and 9, which do not have the same variable parts, are not like terms and cannot be combined directly.

The process of adding polynomials involves several key steps. First, identify the like terms within the polynomials being added. Second, rearrange the terms so that like terms are grouped together. This step is crucial for clarity and helps prevent errors. Third, combine the like terms by adding their coefficients. The variable part of the term remains unchanged. Finally, write the resulting polynomial in a simplified form. By following these steps, one can systematically add polynomials of any complexity. In the next sections, we will apply these steps to the specific example provided, demonstrating each stage of the process in detail.

Step-by-Step Solution

To solve the problem (2a² + ab + 2b) + (4a² - 3ab + 9), we will follow a step-by-step approach:

Identify Like Terms: In this step, we need to identify the terms that have the same variables raised to the same powers. This is the cornerstone of polynomial addition, as it dictates which terms can be combined. Looking at the expression (2a² + ab + 2b) + (4a² - 3ab + 9), we can identify the following pairs of like terms: 2a² and 4a², which both have the variable 'a' raised to the power of 2; ab and -3ab, which both have the variables 'a' and 'b' raised to the power of 1; and the constant terms 2b and 9. It is crucial to recognize that 2b and 9 are not like terms because 2b has the variable 'b' while 9 is a constant without any variable. Similarly, terms like 2a² and ab are not like terms because they have different variable parts (a² and ab, respectively). The ability to accurately identify like terms is fundamental to simplifying polynomial expressions and is a skill that is built upon throughout algebra. By carefully examining each term and comparing their variable parts, we can ensure that we are combining the correct terms in the subsequent steps. This initial step sets the stage for the rest of the solution, so accuracy and attention to detail are paramount.
Rearrange the Expression: The next step involves rearranging the expression to group the like terms together. This is done to visually organize the terms and make it easier to combine them in the following step. By regrouping like terms, we reduce the chance of making errors and enhance clarity in the process. Starting with the original expression (2a² + ab + 2b) + (4a² - 3ab + 9), we can rearrange the terms as follows: 2a² + 4a² + ab - 3ab + 2b + 9. Here, we have brought the terms with a² together (2a² and 4a²), followed by the terms with ab (ab and -3ab), then the term with b (2b), and finally the constant term (9). This rearrangement does not change the value of the expression, as we are simply changing the order of addition, which is a commutative operation. The rearranged expression clearly shows the like terms grouped together, making it straightforward to perform the addition. This step is particularly helpful when dealing with more complex polynomials that have a larger number of terms. It provides a structured approach that simplifies the addition process and minimizes the risk of overlooking any terms. Efficient rearrangement is a key skill in polynomial manipulation and contributes to a more organized and accurate solution.
Combine Like Terms: Now, we combine the like terms by adding their coefficients. This is the core of the polynomial addition process, where we simplify the expression by performing the arithmetic operations on the coefficients of like terms. Taking the rearranged expression 2a² + 4a² + ab - 3ab + 2b + 9, we proceed by adding the coefficients of the like terms. For the terms 2a² and 4a², we add their coefficients 2 and 4, which gives us 6a². For the terms ab and -3ab, we add their coefficients 1 (the implicit coefficient of ab) and -3, resulting in -2ab. The terms 2b and 9 do not have any like terms to combine with, so they remain unchanged. Therefore, the combination of like terms results in the expression 6a² - 2ab + 2b + 9. This step effectively reduces the complexity of the polynomial expression by consolidating terms that share the same variable parts. By accurately adding the coefficients, we ensure that the simplified expression is equivalent to the original one. The process of combining like terms is a fundamental skill in algebra and is used extensively in various mathematical contexts, including solving equations, simplifying expressions, and performing other algebraic operations. Attention to detail and a solid understanding of arithmetic operations are crucial for success in this step.

Final Result and Answer

After combining like terms, we arrive at the simplified polynomial: 6a² - 2ab + 2b + 9. This is the sum of the two original polynomials, (2a² + ab + 2b) and (4a² - 3ab + 9).

Comparing our result with the provided options:

A. 6a⁴ - ab + 2b + 9 B. 6a² - 2ab + 9 C. 6a² + 2ab + 2b + 9 D. 6a² - 2ab + 2b + 9

We can see that option D, 6a² - 2ab + 2b + 9, matches our calculated result. Therefore, the correct answer is D.

The final result of adding the polynomials (2a² + ab + 2b) and (4a² - 3ab + 9) is the polynomial 6a² - 2ab + 2b + 9. This result is obtained by carefully identifying and combining like terms, which is a fundamental skill in algebra. When we compare this result to the given options, we find that it matches option D exactly. The other options contain errors in either the coefficients or the exponents of the variables. Option A, for example, has a term with a⁴, which is incorrect since there were no a⁴ terms in the original polynomials. Option B is missing the term 2b, and option C has the wrong sign for the term with ab. Therefore, the correct answer, option D, is the only one that accurately represents the sum of the two given polynomials. This exercise demonstrates the importance of paying close attention to detail and following the correct steps when adding polynomials. By mastering these skills, students can build a strong foundation in algebra and tackle more complex mathematical problems with confidence.

Conclusion

In conclusion, the sum of the polynomials (2a² + ab + 2b) + (4a² - 3ab + 9) is 6a² - 2ab + 2b + 9. This is achieved by identifying and combining like terms. Polynomial addition is a crucial skill in algebra and understanding this process is essential for further mathematical studies. Understanding how to add polynomials is not just about getting the correct answer; it is about developing a solid foundation in algebraic manipulation. The ability to identify like terms, rearrange expressions, and combine terms efficiently is crucial for success in more advanced topics such as polynomial multiplication, division, and factoring. Moreover, the principles of polynomial addition extend to other areas of mathematics, including calculus and linear algebra. The step-by-step approach outlined in this guide provides a clear and systematic method for adding polynomials, which can be applied to a wide range of problems. By practicing these techniques, students can improve their algebraic skills and build confidence in their ability to solve mathematical problems. The importance of accuracy and attention to detail cannot be overstated, as even a small error in identifying or combining like terms can lead to an incorrect result. Therefore, mastering the fundamentals of polynomial addition is a key step in developing mathematical proficiency and achieving academic success.