Polynomial Division Explained Find The Quotient Of (2x³ - 13x² + 16x - 5) ÷ (x - 5)

Introduction: Mastering Polynomial Division

In the realm of algebra, polynomial division stands as a fundamental operation, much like long division in arithmetic. It allows us to break down complex polynomial expressions into simpler forms, revealing their underlying structure and relationships. Understanding polynomial division is crucial for solving equations, factoring polynomials, and grasping various concepts in advanced mathematics. In this comprehensive guide, we will delve into the process of polynomial division, specifically focusing on the expression (2x³ - 13x² + 16x - 5) ÷ (x - 5). We'll explore the steps involved, providing clear explanations and illustrative examples to ensure a thorough understanding. So, if you're ready to unravel the intricacies of polynomial division, let's embark on this mathematical journey together.

The Essence of Polynomial Division

Before we dive into the specifics of our problem, let's first grasp the essence of polynomial division. Polynomial division is the process of dividing one polynomial (the dividend) by another (the divisor), resulting in a quotient and a remainder. The quotient represents the result of the division, while the remainder is the portion that is "left over" after the division is complete. This process mirrors the long division we perform with numbers, but instead of digits, we're dealing with terms containing variables and exponents. The key to mastering polynomial division lies in understanding the systematic steps involved and applying them with precision. We'll explore these steps in detail as we tackle our main problem, providing a solid foundation for handling any polynomial division scenario. Remember, practice is paramount when it comes to mastering mathematical concepts, so be sure to work through additional examples to solidify your understanding.

Setting the Stage: Understanding the Problem

Our primary task is to determine the quotient when the polynomial 2x³ - 13x² + 16x - 5 is divided by x - 5. This can be represented mathematically as (2x³ - 13x² + 16x - 5) ÷ (x - 5). To solve this, we will employ the method of polynomial long division, a systematic approach that mirrors the long division process used with numerical values. Before we begin, let's identify the key components of our problem: the dividend (2x³ - 13x² + 16x - 5) and the divisor (x - 5). Understanding these components is crucial for setting up the long division problem correctly. Next, we'll need to arrange the terms of the dividend and divisor in descending order of their exponents, ensuring that no powers are skipped. This step is essential for maintaining the correct alignment during the division process. With the problem set up correctly, we're ready to embark on the step-by-step process of polynomial long division, which we'll explore in the following sections.

Step-by-Step Solution: Unveiling the Quotient

Step 1: Setting Up the Long Division

The initial step in polynomial long division is to set up the problem in a format that mirrors traditional long division. We write the dividend, 2x³ - 13x² + 16x - 5, inside the long division symbol, and the divisor, x - 5, outside. It's crucial to ensure that the terms are arranged in descending order of their exponents, which they already are in this case. This setup provides a visual framework for the division process, making it easier to track the steps and avoid errors. The setup also helps us to identify the leading terms of both the dividend and the divisor, which are essential for the subsequent steps. With the problem set up correctly, we're ready to begin the actual division process. This involves a series of steps where we divide, multiply, subtract, and bring down terms, much like in traditional long division. However, instead of dealing with numerical digits, we're working with algebraic terms containing variables and exponents. So, let's move on to the next step and see how the division unfolds.

Step 2: Dividing the Leading Terms

The heart of polynomial long division lies in systematically dividing the leading terms. We begin by focusing on the leading term of the dividend, which is 2x³, and the leading term of the divisor, which is x. We ask ourselves: what do we need to multiply x by to get 2x³? The answer is 2x². This 2x² becomes the first term of our quotient, which we write above the long division symbol, aligning it with the x² term in the dividend. This step is crucial as it sets the foundation for the rest of the division process. By focusing on the leading terms, we ensure that we're systematically reducing the degree of the dividend until we reach a remainder that has a lower degree than the divisor. This process mirrors the way we divide numbers in long division, where we focus on the leading digits to determine the quotient. With the first term of the quotient determined, we move on to the next step, which involves multiplying the divisor by this term.

Step 3: Multiplying the Divisor

Having determined the first term of the quotient, 2x², the next step is to multiply the entire divisor, (x - 5), by this term. This gives us 2x² * (x - 5) = 2x³ - 10x². This result is crucial because we will subtract it from the dividend in the next step. The multiplication process ensures that we account for all the terms in the divisor when determining the portion of the dividend that is "divided out" by the quotient term. This step is analogous to multiplying in traditional long division, where we multiply the quotient digit by the entire divisor. The key here is to distribute the quotient term correctly to each term in the divisor, paying close attention to the signs. A mistake in this step can lead to errors in the subsequent steps and ultimately affect the final quotient and remainder. With the multiplication complete, we're ready to move on to the subtraction step, where we'll see how much of the dividend has been accounted for by our first quotient term.

Step 4: Subtracting and Bringing Down

Now comes the crucial step of subtracting the result from the previous step, 2x³ - 10x², from the corresponding terms in the dividend, 2x³ - 13x². This gives us (2x³ - 13x²) - (2x³ - 10x²) = -3x². We then bring down the next term from the dividend, which is +16x, resulting in the new expression -3x² + 16x. This process of subtraction and bringing down is the heart of long division, allowing us to systematically reduce the dividend until we reach the remainder. The subtraction step effectively cancels out the leading term of the dividend (in this case, 2x³), leaving us with a new expression of a lower degree. Bringing down the next term ensures that we're working with the appropriate portion of the dividend in each step. This process mirrors the way we subtract and bring down digits in traditional long division. With the new expression obtained, we repeat the process, focusing on the leading term to determine the next term in the quotient.

Step 5: Repeating the Process

With the new expression -3x² + 16x, we repeat the division process. We focus on the leading term, -3x², and ask: what do we need to multiply x (the leading term of the divisor) by to get -3x²? The answer is -3x. This becomes the next term in our quotient, which we write above the long division symbol, aligning it with the x term in the dividend. Next, we multiply the divisor, (x - 5), by -3x, which gives us -3x * (x - 5) = -3x² + 15x. We then subtract this result from our current expression: (-3x² + 16x) - (-3x² + 15x) = x. Finally, we bring down the last term from the dividend, which is -5, giving us the new expression x - 5. This iterative process of dividing, multiplying, subtracting, and bringing down is the essence of polynomial long division. Each iteration reduces the degree of the expression we're working with, bringing us closer to the final quotient and remainder. With the process repeated, we move on to the final steps to complete the division.

Step 6: The Final Division

We now have the expression x - 5. We repeat the process one last time. What do we need to multiply x by to get x? The answer is 1. So, 1 becomes the last term in our quotient. Multiplying the divisor, (x - 5), by 1 gives us 1 * (x - 5) = x - 5. Subtracting this from our current expression, we get (x - 5) - (x - 5) = 0. This indicates that we have reached a remainder of 0, meaning the division is complete. The fact that the remainder is zero signifies that the divisor, (x - 5), divides the dividend, (2x³ - 13x² + 16x - 5), evenly. This is analogous to dividing numbers where the remainder is zero, indicating a perfect division. With the remainder at zero, we have successfully completed the polynomial long division, and we can now state the quotient.

The Grand Finale: Presenting the Quotient

Having meticulously navigated the steps of polynomial long division, we have arrived at the quotient. By systematically dividing, multiplying, subtracting, and bringing down terms, we have successfully broken down the complex polynomial expression. Our calculations reveal that the quotient of (2x³ - 13x² + 16x - 5) ÷ (x - 5) is 2x² - 3x + 1. This result represents the polynomial that, when multiplied by the divisor (x - 5), yields the original dividend (2x³ - 13x² + 16x - 5). In essence, the quotient is the answer to our division problem, providing a concise representation of the relationship between the dividend and the divisor. With the quotient determined, we have successfully solved the problem, demonstrating our mastery of polynomial long division. But the journey doesn't end here. Understanding the significance of the quotient and its implications is the next step in solidifying our understanding of polynomial division.

Conclusion: The Significance of the Quotient

In conclusion, we have successfully determined that the quotient of (2x³ - 13x² + 16x - 5) ÷ (x - 5) is 2x² - 3x + 1. This exercise not only demonstrates the mechanics of polynomial long division but also highlights the importance of this technique in simplifying complex algebraic expressions. The quotient 2x² - 3x + 1 represents a polynomial that, when multiplied by (x - 5), gives us the original polynomial 2x³ - 13x² + 16x - 5. This understanding is crucial in various mathematical contexts, including factoring polynomials, solving equations, and analyzing polynomial functions. Polynomial division is a cornerstone of algebraic manipulation, enabling us to break down complex problems into manageable steps. By mastering this technique, we gain a powerful tool for tackling a wide range of mathematical challenges. So, the next time you encounter a polynomial division problem, remember the systematic steps we've explored, and you'll be well-equipped to find the quotient with confidence.

The correct answer is A. 2x² - 3x + 1.