Unveiling The Product Of (-2d^2 + S)(5d^2 - 6s) A Step-by-Step Guide

Understanding the Algebraic Expression

In this comprehensive exploration, we will delve into the intricacies of the algebraic expression (-2d^2 + s)(5d^2 - 6s). Our aim is to dissect this expression, understand its components, and ultimately, unveil the product it represents. This involves expanding the expression, simplifying it, and analyzing the resulting terms. To begin, let's break down the expression into its fundamental parts. We have two binomials: (-2d^2 + s) and (5d^2 - 6s). Each binomial consists of two terms. The first binomial has the terms -2d^2 and s, while the second binomial has the terms 5d^2 and -6s. The expression signifies the multiplication of these two binomials. Understanding the structure of this expression is crucial for effectively expanding and simplifying it. Before we proceed with the expansion, it's important to recognize the variables and coefficients involved. The variables in this expression are 'd' and 's', and they represent unknown quantities. The coefficients are the numerical values that multiply these variables. For example, in the term -2d^2, -2 is the coefficient and d^2 is the variable part. Similarly, in the term 5d^2, 5 is the coefficient. Recognizing these components is essential for performing the multiplication correctly. Now, let's discuss the method we will use to expand this expression. The most common method for multiplying two binomials is the FOIL method, which stands for First, Outer, Inner, Last. This method ensures that each term in the first binomial is multiplied by each term in the second binomial. We will apply this method step-by-step to ensure accuracy and clarity. First, we multiply the first terms of each binomial: (-2d^2) * (5d^2). Next, we multiply the outer terms: (-2d^2) * (-6s). Then, we multiply the inner terms: (s) * (5d^2). Finally, we multiply the last terms: (s) * (-6s). After performing these multiplications, we will combine the resulting terms and simplify the expression. This will give us the final product, which is a polynomial expression in terms of 'd' and 's'. This detailed approach will enable us to fully understand and unveil the product of the given expression.

Expanding the Expression: A Step-by-Step Approach

To expand the expression (-2d^2 + s)(5d^2 - 6s), we will meticulously apply the FOIL method, ensuring each term is correctly multiplied and accounted for. The FOIL method, as previously mentioned, stands for First, Outer, Inner, Last, and it provides a systematic way to multiply two binomials. Let's begin with the First terms. We multiply the first terms of each binomial: (-2d^2) * (5d^2). This gives us -10d^4. Remember that when multiplying variables with exponents, we add the exponents. In this case, d^2 multiplied by d^2 results in d^(2+2) = d^4. The coefficient -2 multiplied by 5 gives us -10. So, the first term of our expanded expression is -10d^4. Next, we move on to the Outer terms. We multiply the outer terms of the expression: (-2d^2) * (-6s). This gives us 12d^2s. Here, we are multiplying -2d^2 by -6s. The coefficients -2 and -6 multiply to give 12. The variables d^2 and s are different, so we simply write them together as d^2s. The positive sign is because a negative times a negative is a positive. Thus, the second term of our expanded expression is 12d^2s. Now, let's consider the Inner terms. We multiply the inner terms of the expression: (s) * (5d^2). This gives us 5d^2s. We are multiplying s by 5d^2. The coefficient is 5, and the variables are s and d^2. We typically write the term with the variable having the higher exponent first, so we write this as 5d^2s. This term is positive because we are multiplying two positive terms. So, the third term of our expanded expression is 5d^2s. Finally, we multiply the Last terms. We multiply the last terms of each binomial: (s) * (-6s). This gives us -6s^2. Here, we are multiplying s by -6s. The coefficient is -6, and the variable s multiplied by s gives us s^2. The term is negative because we are multiplying a positive term by a negative term. Therefore, the fourth term of our expanded expression is -6s^2. Now that we have multiplied all the terms using the FOIL method, we have the following terms: -10d^4, 12d^2s, 5d^2s, and -6s^2. The next step is to combine any like terms to simplify the expression further. This systematic approach ensures that we have correctly expanded the expression, setting the stage for further simplification and analysis.

Simplifying the Expanded Expression: Combining Like Terms

After expanding the expression using the FOIL method, we obtained the terms -10d^4, 12d^2s, 5d^2s, and -6s^2. The next crucial step is to simplify the expanded expression by combining like terms. Like terms are terms that have the same variables raised to the same powers. In our expanded expression, we have two terms that are considered like terms: 12d^2s and 5d^2s. Both of these terms have the variables d^2 and s, and both variables have the same exponents (2 for d and 1 for s). To combine like terms, we simply add their coefficients. In this case, we add the coefficients of 12d^2s and 5d^2s, which are 12 and 5 respectively. Adding these coefficients, we get 12 + 5 = 17. Therefore, the combined term is 17d^2s. Now, let's rewrite the expression with the combined like terms. We have -10d^4, 17d^2s, and -6s^2. The terms -10d^4 and -6s^2 do not have any like terms in the expression, so they remain as they are. Thus, the simplified expression is -10d^4 + 17d^2s - 6s^2. This simplified expression is the final product of the original expression (-2d^2 + s)(5d^2 - 6s). By combining like terms, we have reduced the expression to its simplest form, making it easier to understand and analyze. This simplified form clearly shows the relationship between the variables d and s, and the coefficients that determine their contribution to the overall value of the expression. The process of simplifying expressions is a fundamental concept in algebra, and it is essential for solving equations, graphing functions, and performing various mathematical operations. Understanding how to identify and combine like terms is a key skill for any student of mathematics. In this specific example, by combining the like terms 12d^2s and 5d^2s, we were able to reduce the expression from four terms to three terms. This not only makes the expression more concise but also provides a clearer picture of the underlying mathematical relationship. The simplified expression, -10d^4 + 17d^2s - 6s^2, is the unveiled product we were seeking. It represents the final result of multiplying the two binomials and simplifying the resulting expression. This thorough simplification process ensures that we have accurately determined the product and have a clear understanding of its components.

The Final Product: Analyzing the Resulting Polynomial

After expanding and simplifying the original expression (-2d^2 + s)(5d^2 - 6s), we have arrived at the final product: -10d^4 + 17d^2s - 6s^2. This expression is a polynomial in two variables, 'd' and 's'. To fully understand this result, let's analyze its components and characteristics. The polynomial consists of three terms: -10d^4, 17d^2s, and -6s^2. Each term has a coefficient and a variable part. The first term, -10d^4, has a coefficient of -10 and a variable part of d^4. This term represents a quartic term in 'd', meaning 'd' is raised to the power of 4. The second term, 17d^2s, has a coefficient of 17 and a variable part of d^2s. This term is a mixed term, involving both 'd' and 's'. The degree of this term is 3 (2 from d^2 and 1 from s). The third term, -6s^2, has a coefficient of -6 and a variable part of s^2. This term represents a quadratic term in 's', meaning 's' is raised to the power of 2. The degree of the polynomial is determined by the highest degree of any of its terms. In this case, the term with the highest degree is -10d^4, which has a degree of 4. Therefore, the polynomial -10d^4 + 17d^2s - 6s^2 is a quartic polynomial. Analyzing the signs of the coefficients can also provide insights into the behavior of the polynomial. The term -10d^4 has a negative coefficient, which means that as the absolute value of 'd' increases, this term will become increasingly negative. The term -6s^2 also has a negative coefficient, indicating a similar behavior with respect to 's'. The term 17d^2s has a positive coefficient, which means that this term will contribute positively to the overall value of the polynomial. Understanding the structure and components of this polynomial is crucial for various mathematical applications, such as solving equations, graphing functions, and modeling real-world phenomena. For example, if we were to graph this polynomial, we would expect to see a curve that reflects the quartic nature of the expression, with possible turning points and intercepts determined by the coefficients and exponents. The final product, -10d^4 + 17d^2s - 6s^2, is the culmination of our step-by-step expansion and simplification process. It represents the complete and simplified result of multiplying the two binomials (-2d^2 + s) and (5d^2 - 6s). This analysis provides a thorough understanding of the expression and its properties.

Conclusion: The Significance of Algebraic Manipulation

In conclusion, through a detailed step-by-step process, we have successfully unveiled the product of the algebraic expression (-2d^2 + s)(5d^2 - 6s). We began by understanding the structure of the expression, then meticulously applied the FOIL method to expand it, and finally, we simplified the result by combining like terms. The final product we obtained is -10d^4 + 17d^2s - 6s^2, a quartic polynomial in two variables, 'd' and 's'. This journey through algebraic manipulation highlights the importance of understanding fundamental concepts such as the FOIL method, combining like terms, and analyzing polynomial expressions. These skills are not only essential for solving mathematical problems but also for developing logical thinking and problem-solving abilities in various fields. The process of expanding and simplifying algebraic expressions is a cornerstone of mathematics. It allows us to transform complex expressions into more manageable forms, making them easier to analyze and use in further calculations. The FOIL method, in particular, is a powerful tool for multiplying binomials, ensuring that every term is correctly accounted for. Combining like terms is crucial for simplifying expressions and reducing them to their most concise form. This not only makes the expressions easier to work with but also provides a clearer picture of the underlying mathematical relationships. The ability to analyze polynomial expressions is also a key skill in mathematics. Understanding the degree of a polynomial, the coefficients of its terms, and the behavior of the expression as the variables change are all important aspects of mathematical analysis. These skills are essential for solving equations, graphing functions, and modeling real-world phenomena. The significance of algebraic manipulation extends beyond the realm of pure mathematics. Algebraic skills are used in various fields, including physics, engineering, computer science, and economics. For example, in physics, algebraic equations are used to describe the motion of objects, the behavior of light and sound, and the interactions of particles. In engineering, algebraic principles are used to design structures, circuits, and systems. In computer science, algebraic concepts are used in algorithm design, data analysis, and cryptography. In economics, algebraic models are used to analyze markets, predict economic trends, and make investment decisions. Therefore, mastering algebraic manipulation is not just about solving mathematical problems; it is about developing a valuable skillset that can be applied in a wide range of contexts. The final product we have obtained, -10d^4 + 17d^2s - 6s^2, is a testament to the power and elegance of algebraic manipulation. By carefully applying the rules and principles of algebra, we have transformed a seemingly complex expression into a clear and understandable form. This process exemplifies the beauty and utility of mathematics in simplifying and understanding the world around us.