Polynomial Multiplication Explained: How To Multiply (5r-4)(r^2-6r+4)

Polynomial multiplication is a fundamental operation in algebra, serving as a cornerstone for more advanced mathematical concepts. It involves the process of multiplying two or more polynomials, which are algebraic expressions consisting of variables and coefficients. Mastering this skill is crucial for success in algebra, calculus, and other related fields. In this comprehensive exploration, we will dissect the multiplication of the binomial (5r-4) and the trinomial (r^2-6r+4), providing a step-by-step guide to understanding the underlying principles and techniques. Our focus will be on the distributive property, a core concept that governs polynomial multiplication. We will also delve into combining like terms, a critical step in simplifying the resulting expression. By the end of this article, you will not only be able to confidently multiply polynomials but also grasp the theoretical underpinnings that make this operation so essential in mathematics. Our journey will begin with a detailed breakdown of the problem, followed by a meticulous application of the distributive property. We will then meticulously combine like terms to arrive at the simplified polynomial. Throughout the process, we will highlight key concepts and provide illustrative examples to enhance your understanding. Whether you are a student grappling with algebra for the first time or a seasoned mathematician seeking a refresher, this article will provide valuable insights and practical skills. Polynomial multiplication is not merely a mechanical process; it is a gateway to understanding the intricate relationships between algebraic expressions. By mastering this skill, you will unlock a deeper appreciation for the beauty and power of mathematics.

Breaking Down the Problem: (5r-4)(r^2-6r+4)

Before diving into the multiplication process, let's carefully examine the problem at hand: (5r-4)(r^2-6r+4). We are tasked with multiplying a binomial, (5r-4), by a trinomial, (r^2-6r+4). A binomial is a polynomial with two terms, while a trinomial has three terms. The key to solving this problem lies in the distributive property, which states that multiplying a sum by a number is the same as multiplying each addend separately by the number and then adding the products. In simpler terms, we need to multiply each term in the first polynomial (5r-4) by each term in the second polynomial (r^2-6r+4). This systematic approach ensures that we account for every possible product, leading to an accurate solution. To visualize this process, we can think of it as a series of smaller multiplications: 5r multiplied by each term in (r^2-6r+4), and -4 multiplied by each term in (r^2-6r+4). This methodical breakdown is crucial for avoiding errors and ensuring a clear understanding of the steps involved. We will meticulously execute each of these multiplications, paying close attention to the signs and exponents. The ultimate goal is to expand the product into a single polynomial expression, where each term is clearly defined. This initial step of breaking down the problem is essential for clarity and organization. It sets the stage for the subsequent application of the distributive property and the simplification process. By understanding the structure of the polynomials and the principles of multiplication, we can approach the problem with confidence and precision. Let's now embark on the journey of applying the distributive property, step by meticulous step, to unravel the intricacies of this polynomial multiplication.

Applying the Distributive Property: A Step-by-Step Guide

The heart of polynomial multiplication lies in the application of the distributive property. This property, a cornerstone of algebra, allows us to multiply a sum by a number by multiplying each addend separately and then adding the products. In the context of our problem, (5r-4)(r^2-6r+4), we will distribute each term of the binomial (5r-4) across the trinomial (r^2-6r+4). This process involves two main steps: first, multiplying 5r by each term in the trinomial, and second, multiplying -4 by each term in the trinomial. Let's begin with the first step: multiplying 5r by (r^2-6r+4). This yields three individual products: 5r * r^2, 5r * -6r, and 5r * 4. Performing these multiplications, we get 5r^3, -30r^2, and 20r, respectively. Now, let's move on to the second step: multiplying -4 by (r^2-6r+4). This again yields three individual products: -4 * r^2, -4 * -6r, and -4 * 4. Performing these multiplications, we get -4r^2, 24r, and -16, respectively. We have now successfully distributed each term of the binomial across the trinomial. The next crucial step is to combine the results of these multiplications into a single expression. This expression will contain all the individual products we have calculated, ready for simplification. The distributive property, when applied meticulously, ensures that every term in the first polynomial is multiplied by every term in the second polynomial. This systematic approach is essential for accuracy and completeness. By breaking down the multiplication into smaller, manageable steps, we minimize the risk of errors and gain a clearer understanding of the process. The result of this distribution is a longer polynomial expression, which we will then simplify by combining like terms. This simplification process is the final step in arriving at the solution. Let's proceed to this crucial stage, where we will consolidate our findings and present the final, simplified polynomial.

Combining Like Terms: Simplifying the Expression

After applying the distributive property, we arrive at a longer polynomial expression that needs simplification. This simplification process hinges on the concept of like terms. Like terms are terms that have the same variable raised to the same power. For instance, 3x^2 and -5x^2 are like terms because they both have the variable x raised to the power of 2. Similarly, 7y and 2y are like terms because they both have the variable y raised to the power of 1. However, 2x^2 and 2x are not like terms because the variable x is raised to different powers. The key to combining like terms is to add or subtract their coefficients while keeping the variable and its exponent the same. In our case, after distributing, we have the following terms: 5r^3, -30r^2, 20r, -4r^2, 24r, and -16. To simplify this expression, we need to identify and combine the like terms. First, we look for terms with r^3. We have only one term, 5r^3, so it remains as is. Next, we look for terms with r^2. We have -30r^2 and -4r^2. Combining these, we get -34r^2. Then, we look for terms with r. We have 20r and 24r. Combining these, we get 44r. Finally, we have the constant term -16, which has no like terms to combine with. Now, we assemble the simplified expression by writing the terms in descending order of their exponents: 5r^3 - 34r^2 + 44r - 16. This is the final, simplified polynomial resulting from the multiplication of (5r-4) and (r^2-6r+4). Combining like terms is a crucial step in polynomial simplification. It allows us to reduce a complex expression to its simplest form, making it easier to understand and work with. This process not only streamlines the expression but also reveals the underlying structure of the polynomial. By mastering the art of combining like terms, we gain a deeper understanding of polynomial algebra and its applications. Let's now summarize our journey and highlight the key takeaways from this exploration.

Solution and Verification: The Final Result

Having meticulously applied the distributive property and diligently combined like terms, we have arrived at the solution to our polynomial multiplication problem: (5r-4)(r^2-6r+4). Our step-by-step journey has led us to the simplified polynomial expression: 5r^3 - 34r^2 + 44r - 16. This is the final result of our calculation. To ensure the accuracy of our solution, it is always prudent to verify our work. One way to verify the result is by substituting a numerical value for the variable r in both the original expression and the simplified expression. If the results match, it provides strong evidence that our simplification is correct. For instance, let's substitute r = 2 into the original expression: (5(2)-4)((2)^2-6(2)+4) = (10-4)(4-12+4) = (6)(-4) = -24. Now, let's substitute r = 2 into the simplified expression: 5(2)^3 - 34(2)^2 + 44(2) - 16 = 5(8) - 34(4) + 88 - 16 = 40 - 136 + 88 - 16 = -24. Since both expressions yield the same result, -24, when r = 2, we can be confident in the accuracy of our solution. Verification is a crucial step in any mathematical problem-solving process. It provides a safety net against errors and reinforces our understanding of the concepts involved. By verifying our solution, we not only confirm its correctness but also deepen our grasp of the underlying principles. The solution, 5r^3 - 34r^2 + 44r - 16, represents the product of the binomial (5r-4) and the trinomial (r^2-6r+4). It is a polynomial of degree 3, also known as a cubic polynomial. This result encapsulates the entire process of polynomial multiplication, from the initial distribution to the final simplification. Let's now recap the key concepts and takeaways from this comprehensive exploration.

Key Takeaways: Mastering Polynomial Multiplication

Throughout this comprehensive exploration of polynomial multiplication, we have delved into the intricacies of multiplying the binomial (5r-4) by the trinomial (r^2-6r+4). Our journey has highlighted several key concepts and techniques that are essential for mastering this fundamental algebraic operation. Firstly, we emphasized the distributive property as the cornerstone of polynomial multiplication. This property allows us to multiply each term of one polynomial by each term of another polynomial, ensuring that we account for all possible products. We demonstrated how to systematically apply the distributive property, breaking down the multiplication into smaller, manageable steps. Secondly, we underscored the importance of combining like terms in simplifying polynomial expressions. Like terms, which have the same variable raised to the same power, can be combined by adding or subtracting their coefficients. This simplification process is crucial for reducing a complex expression to its simplest form. Thirdly, we highlighted the significance of verification in ensuring the accuracy of our solutions. By substituting numerical values into both the original and simplified expressions, we can confirm the correctness of our work. This step provides a safety net against errors and reinforces our understanding of the concepts. The solution to our problem, 5r^3 - 34r^2 + 44r - 16, exemplifies the result of meticulous application of the distributive property and diligent combination of like terms. This polynomial, a cubic polynomial, represents the product of the binomial and the trinomial we started with. In conclusion, mastering polynomial multiplication is not merely about memorizing steps; it is about understanding the underlying principles and applying them systematically. The distributive property, combining like terms, and verification are the cornerstones of this skill. By grasping these concepts and practicing diligently, you can confidently tackle any polynomial multiplication problem. This skill is not only essential for success in algebra but also serves as a foundation for more advanced mathematical concepts. Polynomial multiplication is a gateway to understanding the intricate relationships between algebraic expressions, unlocking a deeper appreciation for the beauty and power of mathematics.

Additional Exercises: Practice Makes Perfect

To solidify your understanding of polynomial multiplication, practice is paramount. The more you practice, the more comfortable and confident you will become with the process. Here are a few additional exercises to help you hone your skills:

1. (2x + 3)(x^2 - 4x + 5)
2. (3y - 1)(2y^2 + 5y - 2)
3. (4z + 2)(z^2 - 3z + 1)
4. (x - 5)(x^2 + 5x + 25)
5. (2a + 1)(4a^2 - 2a + 1)

For each of these problems, follow the same steps we outlined in this article: apply the distributive property, combine like terms, and verify your solution. Remember, the key to success is a systematic approach and attention to detail. As you work through these exercises, pay close attention to the signs and exponents. A small error in these areas can lead to an incorrect solution. Don't be discouraged if you encounter difficulties; mistakes are a natural part of the learning process. Analyze your errors, identify the source of the problem, and try again. With consistent practice, you will develop a strong command of polynomial multiplication. Furthermore, consider exploring additional resources, such as textbooks, online tutorials, and practice problems. There are countless resources available to help you deepen your understanding of algebra and polynomial operations. Collaborate with classmates or seek guidance from a teacher or tutor if you encounter persistent challenges. Learning is a collaborative process, and sharing ideas and insights can be invaluable. Embrace the challenge of polynomial multiplication, and you will unlock a powerful tool for solving a wide range of mathematical problems. The skills you develop in this area will serve you well in future mathematical endeavors. So, grab a pencil, paper, and calculator, and embark on the journey of mastering polynomial multiplication. Your efforts will be rewarded with a deeper understanding of algebra and its applications.

What is the product of the expression $(5r-4)(r^2-6r+4)$?

Polynomial Multiplication Explained How to Multiply (5r-4)(r^2-6r+4)