Multiply And Simplify Polynomial Expressions A Step By Step Guide

In this article, we will delve into the process of multiplying a polynomial expression by a monomial and simplifying the result. Specifically, we will address the expression:

$(\left(7 n^3-\frac{28}{9} n^2-\frac{14}{5} n\right)\left(\frac{2}{7} n^2\right)$.

We will meticulously walk through each step, ensuring a clear understanding of the underlying principles and techniques involved. This comprehensive guide aims to equip you with the skills necessary to confidently tackle similar algebraic manipulations.

Understanding the Distributive Property

At the heart of multiplying a polynomial by a monomial lies the distributive property. This fundamental principle states that for any numbers a, b, and c:

a( b + c) = a b + a c

In simpler terms, when we multiply a single term (like our monomial) by an expression containing multiple terms (like our polynomial), we need to distribute the multiplication across each term within the expression. This means we multiply the monomial by the first term of the polynomial, then by the second term, and so on, adding the results together. Applying the distributive property is crucial for accurately expanding and simplifying algebraic expressions. It ensures that each term within the parentheses is properly accounted for in the multiplication process. By understanding and correctly utilizing the distributive property, we can effectively handle complex polynomial multiplications and arrive at the simplified form of the expression. This principle forms the basis for many algebraic manipulations and is a cornerstone of polynomial arithmetic. To illustrate further, consider a scenario where you have a monomial, such as 3x, and a polynomial, such as 2x^2 + 5x - 1. Applying the distributive property would involve multiplying 3x by each term within the polynomial: (3x * 2x^2) + (3x * 5x) + (3x * -1). This results in 6x^3 + 15x^2 - 3x, which is the expanded form of the original expression. The distributive property effectively breaks down a complex multiplication into a series of simpler multiplications, making it easier to manage and solve. In essence, it is a powerful tool for simplifying algebraic expressions and is fundamental to understanding more advanced algebraic concepts. Mastering the distributive property is essential for anyone seeking proficiency in algebra and related mathematical fields. It provides a systematic approach to handling polynomial multiplications, ensuring accuracy and efficiency in the process. Furthermore, the distributive property is not limited to polynomials; it can be applied to various mathematical contexts, making it a versatile and widely applicable principle. Whether dealing with simple expressions or more complex equations, the distributive property remains a reliable method for expanding and simplifying mathematical expressions.

Applying the Distributive Property to Our Expression

Now, let's apply the distributive property to our given expression:

$(\left(7 n^3-\frac{28}{9} n^2-\frac{14}{5} n\right)\left(\frac{2}{7} n^2\right)$

We will multiply the monomial $\frac{2}{7} n^2$ by each term inside the parentheses:

$\frac{2}{7} n^2 * (7 n^3) - \frac{2}{7} n^2 * (\frac{28}{9} n^2) - \frac{2}{7} n^2 * (\frac{14}{5} n)$

This step is crucial as it expands the original expression into a series of simpler multiplications. By distributing the monomial across each term of the polynomial, we set the stage for subsequent simplification. Each term now involves the product of a constant and a variable raised to a power, making it easier to combine like terms and arrive at the final simplified expression. The distributive property acts as a bridge, transforming a complex expression into a manageable form. It is a fundamental tool in algebraic manipulation, allowing us to break down intricate problems into smaller, more solvable parts. Without the distributive property, multiplying polynomials would be a significantly more challenging task. It provides a systematic approach, ensuring that each term is accounted for and properly multiplied. In our specific example, distributing $\frac{2}{7} n^2$ across the polynomial terms allows us to address each multiplication individually, simplifying the overall process. This step-by-step approach is characteristic of algebraic problem-solving, where complex operations are broken down into simpler, more manageable steps. The distributive property is not merely a mathematical rule; it is a strategic tool that enhances our ability to manipulate and simplify algebraic expressions. Its application in this context demonstrates its power and versatility in handling polynomial multiplications. By mastering the distributive property, we gain a valuable skill that is essential for success in algebra and related mathematical fields.

Performing the Multiplication

Next, we perform each multiplication separately:

• $\frac{2}{7} n^2 * 7 n^3 = \frac{2}{7} * 7 * n^2 * n^3 = 2 n^{2+3} = 2 n^5$
• $\frac{2}{7} n^2 * \frac{28}{9} n^2 = \frac{2}{7} * \frac{28}{9} * n^2 * n^2 = \frac{2 * 28}{7 * 9} n^{2+2} = \frac{56}{63} n^4 = \frac{8}{9} n^4$
• $\frac{2}{7} n^2 * \frac{14}{5} n = \frac{2}{7} * \frac{14}{5} * n^2 * n = \frac{2 * 14}{7 * 5} n^{2+1} = \frac{28}{35} n^3 = \frac{4}{5} n^3$

Each of these multiplications involves combining coefficients and adding the exponents of the variable n. Remember that when multiplying terms with the same base, we add their exponents. This is a fundamental rule of exponents that simplifies the multiplication process. By performing each multiplication individually, we break down the problem into smaller, more manageable parts. This approach not only makes the calculations easier but also reduces the likelihood of errors. The simplification of fractions is also a crucial step in this process. For instance, ${\frac{56}{63}}$ simplifies to ${\frac{8}{9}}$, and ${\frac{28}{35}}$ simplifies to ${\frac{4}{5}}$. These simplifications ensure that the final expression is in its most reduced form. The process of multiplying and simplifying terms often involves a combination of numerical and algebraic manipulations. In this case, we are dealing with fractions, exponents, and variables, all of which require careful attention to detail. The ability to accurately perform these operations is essential for success in algebra and related fields. By systematically working through each multiplication, we can arrive at the correct result and gain a deeper understanding of the underlying principles involved. This step-by-step approach is a hallmark of effective problem-solving in mathematics, where complex problems are broken down into simpler, more manageable steps. In summary, the multiplication step involves a combination of numerical calculations, exponent manipulation, and fraction simplification, all of which are essential skills for algebraic problem-solving.

Combining the Results

Now, let's combine the results of the multiplications:

$2 n^5 - \frac{8}{9} n^4 - \frac{4}{5} n^3$

This expression represents the simplified form of the original expression. We have successfully multiplied the monomial by the polynomial and combined the resulting terms. This final step brings together all the individual calculations and simplifications, resulting in a single, concise expression. The order of terms in the final expression is typically arranged in descending order of exponents, which is a standard convention in algebraic notation. This convention makes it easier to compare and manipulate expressions. In our case, the terms are arranged as ${2 n^5 - \frac{8}{9} n^4 - \frac{4}{5} n^3}$, where the exponents decrease from 5 to 4 to 3. The coefficients of each term are also important, as they determine the magnitude of each term's contribution to the overall expression. In this case, the coefficients are 2, -${\frac{8}{9}}$, and -${\frac{4}{5}}$. These coefficients, along with the exponents, define the characteristics of the polynomial. The final expression represents a polynomial of degree 5, as the highest exponent of the variable n is 5. Polynomials are fundamental objects in algebra and have wide-ranging applications in mathematics and other fields. Understanding how to manipulate and simplify polynomials is essential for solving equations, modeling real-world phenomena, and more. The process of simplifying expressions, like the one we have just completed, is a core skill in algebra. It involves applying various rules and properties to reduce an expression to its simplest form. This not only makes the expression easier to understand but also facilitates further calculations and manipulations. In summary, combining the results of the multiplications involves arranging the terms in a standard order, considering the coefficients and exponents, and recognizing the overall characteristics of the resulting polynomial. This final step represents the culmination of the simplification process and provides a concise and manageable expression.

Final Answer

Therefore, the simplified expression is:

$2 n^5 - \frac{8}{9} n^4 - \frac{4}{5} n^3$

This is our final answer, representing the product of the given monomial and polynomial in its simplest form. We have successfully navigated through the process of distributing, multiplying, and simplifying, arriving at a clear and concise result. The final answer encapsulates all the steps taken, from the initial application of the distributive property to the final combination of terms. It is a testament to the power of algebraic manipulation and the importance of understanding fundamental principles. This simplified expression can now be used for further calculations, such as solving equations or evaluating the expression for specific values of n. The ability to simplify algebraic expressions is a crucial skill in mathematics, allowing us to tackle more complex problems and gain deeper insights into mathematical relationships. The process of arriving at the final answer involves a combination of arithmetic, algebraic, and logical reasoning. Each step builds upon the previous one, leading to a coherent and well-structured solution. The final answer is not just a numerical result; it is a representation of the underlying mathematical relationships and the steps taken to uncover them. In summary, the final answer is the culmination of the entire simplification process, representing the product of the given monomial and polynomial in its simplest form. It is a testament to the power of algebraic manipulation and the importance of understanding fundamental principles. By arriving at this final answer, we have demonstrated a mastery of the concepts and techniques involved in multiplying and simplifying polynomial expressions.

Conclusion

In this article, we have demonstrated how to multiply and simplify a polynomial expression by a monomial. The key steps include applying the distributive property, performing the multiplication of terms, and combining like terms. This process is fundamental in algebra and is used extensively in various mathematical contexts. By mastering these techniques, you can confidently tackle more complex algebraic problems and gain a deeper understanding of mathematical relationships. The ability to multiply and simplify expressions is not just a theoretical exercise; it has practical applications in various fields, including engineering, physics, and computer science. Whether you are solving equations, modeling real-world phenomena, or developing algorithms, the skills you have learned in this article will be invaluable. The process of simplification is a cornerstone of mathematical problem-solving, allowing us to break down complex problems into manageable steps. By mastering these steps, we can approach mathematical challenges with confidence and achieve accurate results. The distributive property, the rules of exponents, and the combination of like terms are all essential tools in the algebraic toolkit. By understanding and applying these tools effectively, we can navigate the world of mathematics with greater ease and success. In conclusion, the ability to multiply and simplify polynomial expressions is a fundamental skill that opens doors to a wide range of mathematical and scientific applications. By practicing and mastering these techniques, you will be well-equipped to tackle future challenges and excel in your mathematical endeavors.