Finding The Quotient Of (x³ + 8) Divided By (x + 2)

Introduction: Delving into Polynomial Division

In the realm of mathematics, polynomial division stands as a cornerstone concept, particularly in algebra. It's a fundamental operation that allows us to break down complex polynomial expressions into simpler, more manageable forms. This article will guide you through the process of finding the quotient when dividing the polynomial (x³ + 8) by (x + 2). We'll explore the underlying principles, step-by-step methods, and ultimately reveal the correct answer from the given options. Understanding polynomial division is crucial for various mathematical applications, including factoring, simplifying rational expressions, and solving equations. This article aims to provide a clear and concise explanation, ensuring that you grasp the concept and can confidently apply it to similar problems. We will discuss various methods to approach this problem, including long division and synthetic division, and also explore the application of the sum of cubes factorization. By the end of this article, you will not only know the answer but also understand the process behind it, enhancing your problem-solving skills in algebra.

Understanding the Problem: A Deep Dive into the Expression

Before diving into the solution, let's dissect the problem at hand. We are tasked with finding the quotient of (x³ + 8) ÷ (x + 2). This means we need to determine what polynomial results from dividing the cubic polynomial x³ + 8 by the linear polynomial x + 2. The expression x³ + 8 is a binomial, specifically a sum of cubes, which is a crucial observation that can simplify our approach. Recognizing this pattern allows us to utilize a specific algebraic identity, which we'll explore later. The divisor, x + 2, is a simple linear expression. Understanding the structure of both the dividend (x³ + 8) and the divisor (x + 2) is paramount for choosing the most efficient method of division. We need to identify the terms, their coefficients, and the degree of each polynomial. The degree of a polynomial is the highest power of the variable, which in this case is 3 for the dividend and 1 for the divisor. This difference in degrees is important in polynomial division. Furthermore, understanding the concept of a quotient is essential. The quotient is the result of a division operation, representing how many times the divisor fits into the dividend. In the context of polynomials, the quotient will be another polynomial, potentially of a lower degree than the dividend. With a clear understanding of the problem's components, we can now explore different methods to find the solution.

Method 1: Long Division of Polynomials

Polynomial long division is a systematic method for dividing polynomials, analogous to the long division method used for numbers. It's a reliable technique that can be applied to any polynomial division problem. To apply long division to our problem, (x³ + 8) ÷ (x + 2), we first set up the division in a similar format to numerical long division. The dividend, x³ + 8, is written inside the division symbol, and the divisor, x + 2, is written outside. It's crucial to include placeholder terms for any missing powers of x in the dividend. In this case, we rewrite x³ + 8 as x³ + 0x² + 0x + 8. This ensures proper alignment during the division process. The next step involves dividing the leading term of the dividend (x³) by the leading term of the divisor (x). This gives us x², which becomes the first term of the quotient. We then multiply the entire divisor (x + 2) by x², resulting in x³ + 2x². This product is subtracted from the dividend. The subtraction yields -2x² + 0x + 8. We then bring down the next term (0x) from the dividend. The process is repeated by dividing the leading term of the new result (-2x²) by the leading term of the divisor (x), giving us -2x. This becomes the next term in the quotient. Multiplying the divisor (x + 2) by -2x gives -2x² - 4x. Subtracting this from -2x² + 0x + 8 results in 4x + 8. Finally, we bring down the last term (8) and repeat the process one more time. Dividing 4x by x gives us 4, which is the final term in the quotient. Multiplying the divisor (x + 2) by 4 gives 4x + 8. Subtracting this from 4x + 8 leaves a remainder of 0. Therefore, the quotient of (x³ + 8) ÷ (x + 2) obtained through long division is x² - 2x + 4. This method provides a clear and structured way to perform polynomial division, ensuring accuracy and understanding.

Method 2: Synthetic Division – A Streamlined Approach

Synthetic division offers a more concise and efficient method for dividing a polynomial by a linear divisor of the form x - a. It's particularly useful when the divisor is a simple binomial. To apply synthetic division to our problem, (x³ + 8) ÷ (x + 2), we first identify the value of a. Since our divisor is x + 2, we rewrite it as x - (-2), so a = -2. We then set up the synthetic division table. We write the coefficients of the dividend, x³ + 0x² + 0x + 8, which are 1, 0, 0, and 8, in a row. We also write the value of a, which is -2, to the left of the table. The process begins by bringing down the first coefficient, 1, to the bottom row. This 1 represents the coefficient of the x² term in the quotient. Next, we multiply -2 by 1 and write the result, -2, under the second coefficient, 0. We then add these two numbers, 0 + (-2), which equals -2. This -2 becomes the coefficient of the x term in the quotient. We repeat the process by multiplying -2 by -2, which equals 4, and write it under the third coefficient, 0. Adding these gives us 0 + 4 = 4. This 4 is the constant term in the quotient. Finally, we multiply -2 by 4, which equals -8, and write it under the last coefficient, 8. Adding these gives us 8 + (-8) = 0. This 0 represents the remainder. The numbers in the bottom row, excluding the last one, represent the coefficients of the quotient. In this case, the quotient is 1x² - 2x + 4, or simply x² - 2x + 4. The last number, 0, confirms that there is no remainder. Synthetic division provides a streamlined and efficient way to perform polynomial division, especially when dealing with linear divisors. It's a valuable tool for simplifying algebraic expressions and solving polynomial equations.

Method 3: Utilizing the Sum of Cubes Factorization

An elegant approach to solving this problem involves recognizing the sum of cubes pattern. The dividend, x³ + 8, can be expressed as x³ + 2³, which fits the formula for the sum of cubes: a³ + b³ = (a + b)(a² - ab + b²). In our case, a = x and b = 2. Applying the sum of cubes formula, we can factor x³ + 8 as (x + 2)(x² - 2x + 4). Now, we are dividing this factored expression by (x + 2). So, the problem becomes [(x + 2)(x² - 2x + 4)] ÷ (x + 2). We can see that the term (x + 2) appears in both the numerator and the denominator. Therefore, we can cancel out the (x + 2) terms, leaving us with x² - 2x + 4. This method provides a direct and efficient way to find the quotient by leveraging the algebraic identity of the sum of cubes. It highlights the importance of recognizing patterns in algebraic expressions. The sum of cubes factorization is a powerful tool that simplifies the division process significantly. This approach not only provides the answer but also reinforces the understanding of algebraic identities and their applications. By recognizing and applying the appropriate factorization, we can avoid the more laborious process of long division or synthetic division. This method underscores the importance of having a strong foundation in algebraic identities for efficient problem-solving.

Determining the Correct Answer: A Comparison of Methods

Having explored three distinct methods – long division, synthetic division, and sum of cubes factorization – we have consistently arrived at the same quotient: x² - 2x + 4. This reinforces the correctness of our solution. Now, let's compare our result with the given options:

A. x² + 4 B. x² - 2x + 4 C. x² - 4 D. x² + 2x + 4

By comparing our calculated quotient, x² - 2x + 4, with the options, we can clearly see that option B, x² - 2x + 4, matches our result. Therefore, option B is the correct answer. The other options are incorrect because they do not match the quotient obtained through the division process. Option A, x² + 4, is missing the -2x term. Option C, x² - 4, is the difference of squares, which is not applicable in this case. Option D, x² + 2x + 4, has the wrong sign for the middle term. The consistent result across multiple methods and the direct comparison with the given options provide strong evidence for the correctness of our solution. This exercise highlights the importance of verifying solutions using different methods to ensure accuracy. By understanding the underlying principles of polynomial division and applying various techniques, we can confidently solve similar problems and avoid common errors.

Conclusion: Mastering Polynomial Division

In conclusion, we have successfully determined the quotient of (x³ + 8) ÷ (x + 2) using three different methods: long division, synthetic division, and sum of cubes factorization. Each method provided a clear pathway to the solution, demonstrating the versatility of algebraic techniques. The consistent result, x² - 2x + 4, obtained across all methods reinforces the accuracy of our answer. By comparing our result with the given options, we confirmed that option B, x² - 2x + 4, is the correct answer. This exploration has not only provided the solution to the specific problem but also enhanced our understanding of polynomial division as a fundamental concept in algebra. We've seen how recognizing patterns, such as the sum of cubes, can simplify the division process. We've also learned the importance of choosing the most efficient method based on the problem's structure. Mastering polynomial division is crucial for various mathematical applications, including solving equations, simplifying expressions, and working with rational functions. This comprehensive guide has equipped you with the knowledge and skills to confidently tackle similar problems and further your understanding of algebra. Remember to practice regularly and explore different approaches to deepen your understanding and problem-solving abilities in mathematics.