Subtracting Polynomials A Step-by-Step Guide To (3u^2-5u+4) - (7u^2+10u+6)

In the realm of mathematics, polynomial subtraction is a fundamental operation that builds upon the principles of algebraic manipulation. This process involves finding the difference between two polynomials, which are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Mastering polynomial subtraction is crucial for solving a wide range of mathematical problems, from simplifying algebraic expressions to solving equations and modeling real-world phenomena. The core concept behind subtracting polynomials lies in the distribution of the negative sign and the subsequent combination of like terms. Like terms are those that have the same variable raised to the same power. For instance, in the expression 3x^2 + 5x - 2x^2 + 1, 3x^2 and -2x^2 are like terms, while 5x is a term on its own, and 1 is a constant term. To subtract one polynomial from another, we first distribute the negative sign to each term of the polynomial being subtracted. This means changing the sign of each term within the parentheses. Once the negative sign is distributed, we combine the like terms by adding or subtracting their coefficients. This process effectively simplifies the expression, resulting in a new polynomial that represents the difference between the original two. Polynomial subtraction is not merely an abstract mathematical exercise; it has practical applications in various fields. For example, in physics, it can be used to calculate the net force acting on an object by subtracting the forces acting in opposite directions. In economics, it can be used to determine the profit or loss of a business by subtracting the expenses from the revenue. In computer graphics, polynomial subtraction can be used to manipulate curves and surfaces. Therefore, a solid understanding of polynomial subtraction is essential for anyone pursuing studies or careers in STEM fields.

The given problem presents a classic scenario of polynomial subtraction. We are tasked with subtracting the polynomial (7u^2 + 10u + 6) from the polynomial (3u^2 - 5u + 4). This means we need to find the difference between these two expressions. To accurately perform this subtraction, we must carefully apply the rules of algebraic manipulation. The process begins with rewriting the expression to explicitly show the subtraction operation: (3u^2 - 5u + 4) - (7u^2 + 10u + 6). The next crucial step involves distributing the negative sign across the terms of the second polynomial, (7u^2 + 10u + 6). This means we multiply each term inside the parentheses by -1, effectively changing their signs. The expression then becomes: (3u^2 - 5u + 4) + (-7u^2 - 10u - 6). This transformation is essential because it allows us to treat the subtraction as an addition problem, which simplifies the subsequent steps. Once the negative sign is distributed, we can proceed to combine like terms. Like terms, as mentioned earlier, are terms that have the same variable raised to the same power. In this case, we have u^2 terms, u terms, and constant terms. We will group these terms together to facilitate the combination process. The grouping of like terms is a visual aid that helps to ensure we don't miss any terms and that we combine them correctly. It's a good practice to underline or highlight like terms to make the process even clearer. By carefully setting up the problem in this way, we pave the way for accurate and efficient subtraction of the polynomials.

Now, let's delve into the step-by-step solution to subtract the polynomials. As established earlier, the problem is: (3u^2 - 5u + 4) - (7u^2 + 10u + 6). Our first step is to distribute the negative sign to the second polynomial: (3u^2 - 5u + 4) + (-7u^2 - 10u - 6). This transformation is crucial because it allows us to treat the subtraction as an addition problem. Next, we identify and group like terms. We have three categories of like terms in this expression: terms with u^2, terms with u, and constant terms. Let's group them together: (3u^2 - 7u^2) + (-5u - 10u) + (4 - 6). By grouping like terms, we make it easier to combine them correctly. Now, we combine the coefficients of the like terms. For the u^2 terms, we have 3u^2 - 7u^2 = -4u^2. For the u terms, we have -5u - 10u = -15u. For the constant terms, we have 4 - 6 = -2. By performing these additions and subtractions, we simplify the expression. Finally, we write the simplified polynomial by combining the results from each group of like terms: -4u^2 - 15u - 2. This is the result of subtracting the polynomial (7u^2 + 10u + 6) from the polynomial (3u^2 - 5u + 4). Each step in this process is crucial for arriving at the correct answer. Distributing the negative sign, grouping like terms, and combining coefficients are all fundamental skills in polynomial manipulation.

To further clarify the solution, let's break down the calculation of each term in detail. This will provide a deeper understanding of how the coefficients are combined during polynomial subtraction. First, consider the u^2 terms. We have 3u^2 from the first polynomial and -7u^2 from the second polynomial (after distributing the negative sign). To combine these terms, we simply add their coefficients: 3 + (-7) = -4. Therefore, the resulting u^2 term is -4u^2. This step involves basic arithmetic, but it's crucial for ensuring the accuracy of the final result. Next, let's look at the u terms. We have -5u from the first polynomial and -10u from the second polynomial (after distributing the negative sign). Again, we add their coefficients: -5 + (-10) = -15. Thus, the resulting u term is -15u. It's important to pay attention to the signs of the coefficients when adding or subtracting. Finally, we consider the constant terms. We have 4 from the first polynomial and -6 from the second polynomial (after distributing the negative sign). Adding these constants gives us: 4 + (-6) = -2. So, the constant term in the resulting polynomial is -2. By examining each term individually, we can see how the coefficients are combined according to the rules of polynomial subtraction. This detailed breakdown reinforces the importance of careful arithmetic and attention to signs. The resulting polynomial, -4u^2 - 15u - 2, is the final answer, representing the difference between the original two polynomials.

In conclusion, after meticulously performing the subtraction, we find that subtracting (7u^2 + 10u + 6) from (3u^2 - 5u + 4) yields the polynomial -4u^2 - 15u - 2. This result is obtained by carefully distributing the negative sign, grouping like terms, and combining their coefficients. Each step in the process is crucial for ensuring accuracy. The final polynomial, -4u^2 - 15u - 2, represents the difference between the two original polynomials. It is a simplified expression that captures the essence of the subtraction operation. Mastering polynomial subtraction is a fundamental skill in algebra, with applications in various fields, including physics, economics, and computer science. A solid understanding of this operation is essential for solving more complex mathematical problems. The process involves distributing the negative sign, identifying and grouping like terms, and combining coefficients. By following these steps carefully, one can confidently subtract polynomials and arrive at the correct result. This problem serves as a valuable exercise in algebraic manipulation, reinforcing the importance of attention to detail and adherence to mathematical rules. The ability to subtract polynomials accurately is a cornerstone of algebraic proficiency, paving the way for more advanced mathematical concepts and applications. Therefore, practicing and mastering this skill is highly beneficial for anyone pursuing studies or careers in STEM fields.