Polynomial Long Division Step-by-Step Solution (x^5 + 18x^2 - 27x) Divided By (x + 3)

In the realm of algebra, polynomial division stands as a fundamental operation, akin to long division with numbers. Mastering this technique opens doors to simplifying complex expressions, solving equations, and gaining deeper insights into the behavior of polynomial functions. This comprehensive guide delves into the process of polynomial long division, using the specific example of dividing the polynomial (x^5 + 18x^2 - 27x) by the binomial (x + 3). Through a meticulous step-by-step approach, we will unravel the intricacies of this method, empowering you to confidently tackle similar problems. Polynomial long division, a cornerstone of algebraic manipulation, allows us to divide a polynomial by another polynomial of equal or lower degree. The process mirrors the familiar long division algorithm used with numbers, but instead of dealing with digits, we manipulate terms containing variables and exponents. The result of polynomial long division is a quotient and a remainder, where the quotient represents the polynomial that results from the division, and the remainder is the polynomial left over after the division is complete. Understanding the mechanics of polynomial long division is crucial for various algebraic tasks, including simplifying rational expressions, factoring polynomials, and solving polynomial equations. The ability to divide polynomials efficiently and accurately lays the foundation for more advanced mathematical concepts and applications.

Setting Up the Long Division

Before embarking on the division process, meticulous preparation is key to ensuring accuracy and clarity. This initial setup lays the groundwork for a smooth and successful execution of the long division algorithm. The first step involves arranging the dividend, which is the polynomial being divided (in our case, x^5 + 18x^2 - 27x), inside the division symbol. It's crucial to write the polynomial in descending order of exponents, ensuring that terms with higher powers of x appear before terms with lower powers. Furthermore, we must account for any missing terms by inserting placeholders with a coefficient of zero. In our example, we notice that the terms x^4 and x^3 are absent. To maintain the proper structure and avoid confusion during the division process, we insert 0x^4 and 0x^3 as placeholders. This ensures that each power of x is represented, even if its coefficient is zero. Now, the dividend appears as x^5 + 0x^4 + 0x^3 + 18x^2 - 27x + 0, where we've also added a placeholder for the constant term. Next, we write the divisor, which is the polynomial we are dividing by (in our case, x + 3), to the left of the division symbol. The divisor remains as it is, without any modifications. With the dividend and divisor properly positioned, we are ready to commence the long division process. This careful setup, with attention to descending exponents and placeholders for missing terms, is paramount for accurate and efficient polynomial division. By ensuring that the terms are arranged correctly, we minimize the chances of errors and streamline the subsequent steps of the algorithm.

Step-by-Step Long Division

With the setup complete, we now embark on the iterative process of long division, systematically working through each step to arrive at the quotient and remainder. This process, while seemingly intricate at first, becomes intuitive with practice. Our first task is to focus on the leading terms of both the dividend (x^5) and the divisor (x). We ask ourselves: "What must we multiply the leading term of the divisor (x) by to obtain the leading term of the dividend (x^5)?" The answer is x^4, since x * x^4 = x^5. We write this x^4 above the division symbol, aligning it with the x^4 term in the dividend. Next, we multiply the entire divisor (x + 3) by the term we just wrote in the quotient (x^4). This gives us x^4 * (x + 3) = x^5 + 3x^4. We write this result below the corresponding terms in the dividend. Now, we subtract the result (x^5 + 3x^4) from the dividend (x^5 + 0x^4). This yields (x^5 + 0x^4) - (x^5 + 3x^4) = -3x^4. We bring down the next term from the dividend, which is 0x^3, and write it next to -3x^4, forming the new expression -3x^4 + 0x^3. We now repeat the process, focusing on the new leading term (-3x^4) and the leading term of the divisor (x). We ask: "What must we multiply x by to obtain -3x^4?" The answer is -3x^3. We write -3x^3 in the quotient, aligning it with the x^3 term. We multiply the divisor (x + 3) by -3x^3, obtaining -3x^3 * (x + 3) = -3x^4 - 9x^3. We write this below the corresponding terms and subtract: (-3x^4 + 0x^3) - (-3x^4 - 9x^3) = 9x^3. We bring down the next term, 18x^2, forming the expression 9x^3 + 18x^2. We continue this iterative process, dividing the leading term (9x^3) by x, obtaining 9x^2. We write 9x^2 in the quotient, multiply the divisor by 9x^2, and subtract. This yields 9x^2 * (x + 3) = 9x^3 + 27x^2, and (9x^3 + 18x^2) - (9x^3 + 27x^2) = -9x^2. We bring down the next term, -27x, forming -9x^2 - 27x. Again, we divide the leading term (-9x^2) by x, obtaining -9x. We write -9x in the quotient, multiply the divisor by -9x, and subtract. This gives us -9x * (x + 3) = -9x^2 - 27x, and (-9x^2 - 27x) - (-9x^2 - 27x) = 0. Finally, we bring down the last term, 0. Since the result of the subtraction is 0, and there are no more terms to bring down, the division is complete. The quotient is the polynomial we wrote above the division symbol: x^4 - 3x^3 + 9x^2 - 9x, and the remainder is 0.

Interpreting the Result

The culmination of the long division process yields a quotient and a remainder, both of which hold significant meaning in the context of polynomial division. Understanding how to interpret these results is crucial for drawing meaningful conclusions and applying them to further algebraic manipulations. In our specific example, where we divided (x^5 + 18x^2 - 27x) by (x + 3), we arrived at a quotient of x^4 - 3x^3 + 9x^2 - 9x and a remainder of 0. The quotient, x^4 - 3x^3 + 9x^2 - 9x, represents the polynomial that results from the division. In simpler terms, it's the answer to the division problem. It tells us that when we divide (x^5 + 18x^2 - 27x) by (x + 3), we obtain x^4 - 3x^3 + 9x^2 - 9x. The remainder, in this case, is 0. A remainder of 0 signifies that the divisor (x + 3) divides the dividend (x^5 + 18x^2 - 27x) evenly. This means that (x + 3) is a factor of (x^5 + 18x^2 - 27x). In other words, we can write (x^5 + 18x^2 - 27x) as the product of (x + 3) and the quotient (x^4 - 3x^3 + 9x^2 - 9x). If the remainder were not 0, it would indicate that the divisor does not divide the dividend evenly. The remainder would then be a polynomial of degree less than the degree of the divisor. The relationship between the dividend, divisor, quotient, and remainder can be expressed by the following equation: Dividend = (Divisor * Quotient) + Remainder. In our case, this translates to (x^5 + 18x^2 - 27x) = (x + 3) * (x^4 - 3x^3 + 9x^2 - 9x) + 0. This equation provides a concise summary of the long division process and highlights the interplay between the different components. Understanding the meaning of the quotient and remainder is essential for various algebraic tasks, such as factoring polynomials, solving equations, and simplifying rational expressions. By correctly interpreting the results of polynomial long division, we gain valuable insights into the relationships between polynomials and their factors.

Choosing the Correct Answer

Having meticulously performed the long division and arrived at the quotient x^4 - 3x^3 + 9x^2 - 9x, we now turn our attention to the multiple-choice options provided. The objective is to identify the option that precisely matches our calculated quotient. This step underscores the importance of accuracy throughout the long division process, as even a minor error can lead to selecting the incorrect answer. We carefully examine each option, comparing it term by term with our result. Option A, x^4 + x^3 - 6x^2 - 9x, differs from our quotient in several terms. The coefficients of the x^3 and x^2 terms do not match, indicating that this option is incorrect. Option B, x^4 - 3x^3 + 9x^2 - 9x, aligns perfectly with our calculated quotient. Each term, along with its coefficient, matches our result exactly. This suggests that option B is the correct answer. Option C, x^4 + 3x^3 - 9x^2 - 9x, deviates from our quotient in the signs of the x^3 and x^2 terms. The coefficients have the opposite signs, making this option incorrect. Option D, x^4 - x^3 + 6x^2 - 9x, also differs from our quotient in multiple terms. The coefficients of the x^3 and x^2 terms do not match, ruling out this option. Through a process of careful comparison, we have confidently identified option B as the correct answer. This highlights the significance of not only performing the long division accurately but also meticulously comparing the result with the given options. By paying close attention to detail, we can ensure that we select the correct answer and demonstrate a thorough understanding of the polynomial long division process. The ability to accurately perform polynomial long division and interpret the results is a fundamental skill in algebra, with applications in various mathematical contexts.

Conclusion

In conclusion, mastering polynomial long division is an essential skill in algebra. Through this detailed guide, we have meticulously walked through the process of dividing (x^5 + 18x^2 - 27x) by (x + 3), demonstrating each step with clarity and precision. From setting up the problem to interpreting the quotient and remainder, we have emphasized the importance of accuracy and attention to detail. The correct quotient, as determined through long division, is x^4 - 3x^3 + 9x^2 - 9x, which corresponds to option B. This exercise not only reinforces the mechanics of polynomial long division but also underscores its significance in simplifying expressions, factoring polynomials, and solving equations. Polynomial long division serves as a foundational technique for more advanced algebraic concepts. By understanding its intricacies and practicing diligently, students can build a strong foundation for future mathematical endeavors. The ability to confidently perform polynomial long division empowers individuals to tackle complex algebraic problems with ease and efficiency. This skill is not only valuable in academic settings but also in various fields that rely on mathematical modeling and analysis. Through consistent practice and a thorough understanding of the underlying principles, anyone can master polynomial long division and unlock its numerous applications.