In this comprehensive guide, we will delve into the world of polynomial division, specifically focusing on the technique of synthetic division. Our primary goal is to solve the expression (x^3 + 1) ÷ (x - 1) using synthetic division and accurately identify the quotient. Synthetic division offers a streamlined method for dividing polynomials, especially when the divisor is a linear expression. This method simplifies the long division process, making it quicker and less prone to errors. Understanding synthetic division is crucial for various mathematical applications, including finding roots of polynomials, simplifying algebraic expressions, and solving complex equations. Throughout this guide, we will break down each step, ensuring a clear and thorough understanding of the process. Whether you are a student grappling with algebra or someone looking to refresh their mathematical skills, this guide will provide you with the knowledge and confidence to tackle polynomial division problems effectively.
Before we dive into solving the specific problem, let's first understand the fundamentals of synthetic division. Synthetic division is a simplified method of dividing a polynomial by a linear divisor of the form (x - a). It is an efficient alternative to long division, particularly when dealing with higher-degree polynomials. The key advantage of synthetic division lies in its streamlined approach, which reduces the complexity of polynomial division to a series of simple arithmetic operations. This not only saves time but also minimizes the chances of making errors. The process involves using only the coefficients of the polynomial and the constant term from the divisor. By setting up the synthetic division table correctly, we can systematically perform the division. Understanding the underlying principles of synthetic division is essential for mastering its application. This section will lay the groundwork for the subsequent steps, ensuring that you grasp the core concepts before we apply them to solve the given problem. By the end of this section, you will have a solid understanding of how synthetic division works and why it is a valuable tool in algebra.
The first step in synthetic division is setting up the problem correctly. This involves extracting the coefficients of the dividend (the polynomial being divided) and determining the value to use from the divisor. In our case, the dividend is x^3 + 1, which can be written as x^3 + 0x^2 + 0x + 1 to include all powers of x. The coefficients are therefore 1, 0, 0, and 1. The divisor is (x - 1), so we take the value 'a' such that x - a = x - 1, which means a = 1. This value will be placed outside the division symbol in the synthetic division setup. Arranging the coefficients and the 'a' value in the correct positions is crucial for the accuracy of the process. A clear and organized setup will make the subsequent steps of synthetic division much easier to follow and execute. This section will guide you through the initial setup, ensuring that you have a solid foundation before moving on to the computational steps. By paying close attention to the details in this setup phase, you will be well-prepared to complete the synthetic division process successfully.
Now, let's perform the synthetic division step by step. Start by bringing down the first coefficient of the dividend, which in our case is 1. Then, multiply this number by the 'a' value (which is also 1) and write the result under the next coefficient (0). Add these two numbers together (0 + 1 = 1) and write the sum below. Repeat this process: multiply the new sum (1) by the 'a' value (1) and write the result under the next coefficient (0). Add these numbers (0 + 1 = 1) and write the sum below. Finally, multiply the latest sum (1) by the 'a' value (1) and write the result under the last coefficient (1). Add these numbers (1 + 1 = 2). The numbers on the bottom row, excluding the last one, represent the coefficients of the quotient, and the last number is the remainder. This iterative process forms the core of synthetic division, allowing us to efficiently determine the quotient and remainder. By carefully following each step, you can accurately perform synthetic division and obtain the correct result. This section will provide a detailed walkthrough of the process, ensuring that you understand how each step contributes to the final solution.
After performing the synthetic division, we need to interpret the results to find the quotient and remainder. The numbers in the bottom row, excluding the last one, are the coefficients of the quotient. In our case, the bottom row (excluding the last number) is 1, 1, and 1. Since we started with a cubic polynomial (x^3), the quotient will be a quadratic polynomial (one degree less). Therefore, the quotient is 1x^2 + 1x + 1, or simply x^2 + x + 1. The last number in the bottom row is the remainder, which in our case is 2. To express the result of the division, we write the quotient plus the remainder divided by the divisor. This gives us x^2 + x + 1 + 2/(x - 1). Understanding how to interpret the results of synthetic division is crucial for accurately solving polynomial division problems. This section will guide you through the process of translating the numerical results into the correct algebraic expression, ensuring that you can confidently identify the quotient and remainder.
The final step is to identify the quotient from the results of the synthetic division. As we determined in the previous section, the coefficients of the quotient are 1, 1, and 1. This corresponds to the polynomial x^2 + x + 1. The remainder, 2, is then expressed as a fraction with the original divisor (x - 1) as the denominator, giving us 2/(x - 1). Therefore, the complete result of the division (x^3 + 1) ÷ (x - 1) is x^2 + x + 1 + 2/(x - 1). Among the given options, this matches option C. Identifying the quotient correctly is the culmination of the synthetic division process. It demonstrates your understanding of the method and your ability to apply it to solve polynomial division problems. This section reinforces the importance of accurately interpreting the results and selecting the correct answer from the given choices.
In conclusion, we have successfully used synthetic division to solve the problem (x^3 + 1) ÷ (x - 1) and identified the quotient as x^2 + x + 1 + 2/(x - 1). This comprehensive guide has walked you through each step of the process, from understanding the fundamentals of synthetic division to setting up the problem, performing the division, interpreting the results, and finally, identifying the quotient. Synthetic division is a powerful tool for dividing polynomials, and mastering this technique is essential for success in algebra and beyond. By following the steps outlined in this guide, you can confidently tackle polynomial division problems and achieve accurate results. Whether you are a student preparing for an exam or someone looking to enhance their mathematical skills, this guide has provided you with the knowledge and understanding necessary to excel in polynomial division. Remember to practice regularly to solidify your skills and build confidence in your ability to solve complex mathematical problems.
Therefore, the correct answer is:
C. x^2 + x + 1 + 2/(x - 1)