Synthetic Division (x^4-1) ÷ (x-1) Find The Quotient

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Synthetic division is a streamlined method for dividing a polynomial by a linear expression of the form x - a. This technique simplifies the long division process, making it faster and more efficient, especially when dealing with higher-degree polynomials. In this comprehensive guide, we will delve into the step-by-step process of using synthetic division to solve the problem (x^4 - 1) ÷ (x - 1) and accurately determine the quotient. We will begin by meticulously setting up the synthetic division, ensuring we account for all coefficients, including those of missing terms. Then, we will walk through the iterative steps of bringing down, multiplying, and adding, which are the core mechanics of this method. Following this detailed walkthrough, we will carefully interpret the result obtained from the synthetic division to identify the quotient. This involves understanding how the numbers in the bottom row translate back into polynomial coefficients and the corresponding degrees of x. Furthermore, we will discuss the significance of the remainder and its implications for the divisibility of the original polynomial. Finally, we will reinforce your understanding with additional examples and practice problems, allowing you to confidently apply synthetic division to a variety of polynomial division scenarios. By mastering synthetic division, you will enhance your algebraic skills and gain a valuable tool for simplifying complex polynomial calculations, which are frequently encountered in various mathematical and scientific contexts. Whether you are a student preparing for an exam or a professional needing to perform quick polynomial divisions, this guide will equip you with the knowledge and confidence to tackle these problems effectively.

Understanding Synthetic Division

Before we dive into the specific problem, let's solidify our understanding of synthetic division. Synthetic division is a simplified way to divide a polynomial by a linear divisor of the form (x - a). It's an efficient alternative to long division, particularly useful for higher-degree polynomials. The key advantage of synthetic division lies in its streamlined approach, which focuses solely on the coefficients of the polynomials involved. This not only reduces the computational complexity but also minimizes the chances of making errors during the division process. By eliminating the explicit use of variables and exponents, synthetic division allows for a more concise and organized method, especially beneficial when dealing with polynomials of degree three or higher. The method's systematic nature makes it easier to track intermediate steps and arrive at the correct quotient and remainder. Synthetic division is not just a computational shortcut; it's also a powerful tool for understanding the relationships between polynomial roots and factors, providing a deeper insight into polynomial behavior. Mastering this technique is crucial for anyone studying algebra and calculus, as it simplifies complex polynomial manipulations and lays the foundation for more advanced mathematical concepts.

Setting up Synthetic Division

To set up synthetic division for the given problem, (x^4 - 1) ÷ (x - 1), we first need to identify the coefficients of the dividend (x^4 - 1) and the root of the divisor (x - 1). The dividend can be rewritten as x^4 + 0x^3 + 0x^2 + 0x - 1. Notice that we've included the terms with zero coefficients for the missing powers of x. This step is crucial in synthetic division, as it ensures that each place value is accounted for, preventing errors in the calculation. The coefficients of the dividend are therefore 1, 0, 0, 0, and -1. The divisor is (x - 1), so the root we'll use is a = 1. Now, we set up the synthetic division table. Write the root (1) to the left, and the coefficients (1, 0, 0, 0, -1) to the right, leaving space below the coefficients for calculations and the resulting quotient. This setup provides a clear and organized framework for performing the synthetic division steps. The root of the divisor is placed outside the division symbol, while the coefficients of the dividend are arranged in a row inside the symbol. This visual arrangement helps to keep track of the numbers and the operations involved, making the process less prone to errors. Proper setup is the foundation of successful synthetic division; without it, even the correct steps can lead to incorrect results. Therefore, taking the time to accurately set up the problem is a worthwhile investment in the overall solution.

Performing the Division

Now, let's execute the steps of synthetic division. The process involves a series of bringing down, multiplying, and adding. First, bring down the leading coefficient (1) to the bottom row. Next, multiply this number (1) by the root (1), and write the result (1) under the next coefficient (0). Add these two numbers (0 + 1 = 1) and write the sum (1) in the bottom row. Repeat this process: multiply the last number in the bottom row (1) by the root (1), write the result (1) under the next coefficient (0), and add (0 + 1 = 1). Continue this pattern until you've processed all coefficients. Multiply the last number in the bottom row (1) by the root (1), write the result (1) under the next coefficient (0), and add (0 + 1 = 1). Finally, multiply the last number in the bottom row (1) by the root (1), write the result (1) under the last coefficient (-1), and add (-1 + 1 = 0). The last number in the bottom row (0) is the remainder, and the other numbers (1, 1, 1, 1) are the coefficients of the quotient. Each step in the synthetic division process builds upon the previous one, creating a chain of calculations that efficiently leads to the solution. The repetitive nature of the steps makes it relatively easy to learn and execute, even for complex polynomial divisions. The key is to maintain accuracy in each multiplication and addition, ensuring that the correct values are carried forward. By following this methodical approach, you can confidently perform synthetic division and obtain the correct quotient and remainder.

Interpreting the Result

After completing the synthetic division, the bottom row of numbers holds the key to the quotient and the remainder. In our case, the bottom row is 1, 1, 1, 1, and 0. The last number (0) represents the remainder. The remaining numbers (1, 1, 1, 1) are the coefficients of the quotient. Since we started with a polynomial of degree 4 (x^4), and we divided by a polynomial of degree 1 (x - 1), the quotient will be a polynomial of degree 3. Therefore, the coefficients 1, 1, 1, and 1 correspond to the terms x^3, x^2, x, and the constant term, respectively. Thus, the quotient is x^3 + x^2 + x + 1. The remainder of 0 indicates that (x - 1) divides evenly into (x^4 - 1). Understanding how to interpret the results of synthetic division is crucial for extracting the desired information from the calculations. The coefficients of the quotient provide the building blocks for the resulting polynomial, while the remainder gives insight into the divisibility of the original polynomial. A remainder of zero signifies that the divisor is a factor of the dividend, which is a fundamental concept in polynomial algebra. By mastering the interpretation of synthetic division results, you can gain a deeper understanding of polynomial relationships and their properties.

The quotient of (x^4-1) egin{align*} \div (x-1) \end{align*} is x3+x2+x+1x^3+x^2+x+1.

Therefore, the correct answer is C. x3+x2+x+1x^3+x^2+x+1.

Additional Examples and Practice

To further solidify your understanding of synthetic division, let's explore additional examples and practice problems. These exercises will help you become more comfortable with the process and build your confidence in applying it to various scenarios. Consider the division (2x^3 - 5x^2 + 3x + 4) ÷ (x - 2). First, set up the synthetic division table with the coefficients 2, -5, 3, and 4, and the root 2. Perform the synthetic division steps: bring down the 2, multiply by 2 to get 4, add to -5 to get -1, multiply by 2 to get -2, add to 3 to get 1, multiply by 2 to get 2, and add to 4 to get 6. The quotient is 2x^2 - x + 1, and the remainder is 6. Another example is (x^4 + 3x^3 - 2x + 5) ÷ (x + 1). Remember to include the 0 coefficient for the missing x^2 term. The coefficients are 1, 3, 0, -2, and 5, and the root is -1. After performing the synthetic division, you'll find the quotient is x^3 + 2x^2 - 2x and the remainder is 7. Practice problems might include dividing x^3 - 8 by x - 2, or dividing 3x^4 + x^3 - 5x^2 + 2x - 1 by x + 1. Working through these examples and practice problems will reinforce the steps of synthetic division and help you develop the skills to handle more complex polynomial divisions. Each problem provides an opportunity to apply the method and verify your understanding, making you proficient in this valuable algebraic technique.

Conclusion

In conclusion, synthetic division is a powerful tool for dividing polynomials by linear expressions. It provides an efficient and organized method to determine the quotient and remainder, simplifying complex algebraic calculations. By understanding the step-by-step process, from setting up the division to interpreting the results, you can confidently apply this technique to a wide range of problems. The ability to perform synthetic division is not only valuable for academic purposes but also for various applications in mathematics, science, and engineering. Mastering this method enhances your problem-solving skills and provides a solid foundation for more advanced mathematical concepts. Through practice and application, you can become proficient in synthetic division and leverage its benefits to tackle polynomial division challenges effectively.