Solving (x^3 + 1) ÷ (x - 1) With Synthetic Division A Step-by-Step Guide

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Synthetic division is a streamlined method for dividing polynomials, particularly useful when the divisor is a linear expression of the form x - a. It offers a more efficient alternative to long division, simplifying the process and reducing the chance of errors. In this comprehensive guide, we will delve into the intricacies of synthetic division, using the example problem (x³ + 1) ÷ (x - 1) to illustrate each step. We will not only provide the solution but also ensure a thorough understanding of the underlying principles, making polynomial division accessible to all.

Polynomial division is a fundamental operation in algebra, enabling us to break down complex polynomial expressions into simpler forms. This skill is crucial for various mathematical applications, including solving equations, factoring polynomials, and graphing functions. Synthetic division, in particular, shines when the divisor is a linear expression, as it allows for a more concise and organized approach compared to long division. Its efficiency and ease of use make it a valuable tool for anyone working with polynomials.

In this article, we will embark on a journey through the world of synthetic division. We will begin by laying the groundwork, defining polynomials and their division. Then, we will dive into the heart of synthetic division, dissecting the process step by step. Using the example problem (x³ + 1) ÷ (x - 1) as our guide, we will demonstrate how to set up the problem, perform the calculations, and interpret the results. By the end of this article, you will not only be able to solve this specific problem but also possess the knowledge and confidence to tackle any polynomial division problem using synthetic division. We will explore the significance of place holders, the mechanics of bringing down and multiplying, and the art of adding and interpreting the final row. This comprehensive approach will equip you with the skills to confidently navigate the realm of polynomial division.

Setting Up Synthetic Division: A Crucial First Step

The first step in synthetic division is to set up the problem correctly. This involves extracting the coefficients of the dividend (x³ + 1) and identifying the root of the divisor (x - 1). Let's break down this process:

  1. Dividend Coefficients: The dividend is the polynomial being divided, in this case, x³ + 1. To extract the coefficients, we need to consider all the powers of x, from the highest to the constant term. It's crucial to include placeholders for any missing terms. Our dividend can be rewritten as 1x³ + 0x² + 0x + 1. Thus, the coefficients are 1, 0, 0, and 1. These coefficients will form the first row in our synthetic division setup.

The importance of placeholders cannot be overstated. Failing to include a placeholder for a missing term will lead to incorrect results. For instance, in this example, omitting the 0x² and 0x terms would disrupt the alignment of the coefficients and skew the final answer. Placeholders ensure that each term is accounted for and that the division process proceeds smoothly.

  1. Divisor Root: The divisor is the polynomial we are dividing by, in this case, x - 1. To find the root, we set the divisor equal to zero and solve for x: x - 1 = 0 => x = 1. This root, 1, will be placed outside the division symbol, acting as the key value in our synthetic division process.

With the coefficients and the root in hand, we are ready to set up the synthetic division table. We write the root (1) to the left, followed by the coefficients (1, 0, 0, 1) in a row. A horizontal line is drawn below the coefficients, and a space is left below the line for the intermediate calculations and the final result. This setup provides a visual framework for the synthetic division process, organizing the numbers and guiding the calculations.

Proper setup is paramount to the success of synthetic division. It lays the foundation for accurate calculations and ultimately leads to the correct quotient and remainder. By meticulously extracting coefficients and identifying the root, we ensure that the synthetic division process unfolds smoothly and efficiently.

Performing Synthetic Division: A Step-by-Step Guide

Now that we have set up the synthetic division, let's walk through the calculations step by step:

  1. Bring Down: The first step is to bring down the leading coefficient (which is 1 in our example) below the horizontal line. This number becomes the first digit of our quotient.

This seemingly simple step is the cornerstone of the entire process. By bringing down the leading coefficient, we initiate the chain of calculations that will ultimately reveal the quotient and remainder. It's a crucial starting point that sets the stage for the subsequent multiplications and additions.

  1. Multiply: Next, we multiply the number we just brought down (1) by the root of the divisor (1). The result (1 * 1 = 1) is written below the next coefficient (0) in the dividend.

This multiplication step is where the magic of synthetic division begins to unfold. We are essentially using the root of the divisor to reverse the process of polynomial multiplication. By multiplying the previous result by the root, we are generating a term that will help us cancel out terms in the original dividend.

  1. Add: Now, we add the numbers in the second column (0 and 1). The sum (0 + 1 = 1) is written below the horizontal line.

This addition step is where we combine the terms generated by the multiplication with the original coefficients of the dividend. The sum represents the coefficient of the next term in the quotient. It's a crucial step in reducing the degree of the polynomial and moving closer to the final answer.

  1. Repeat: We repeat steps 2 and 3 for the remaining columns. Multiply the latest number below the line (1) by the root (1) and write the result (1 * 1 = 1) below the next coefficient (0). Add the numbers in the third column (0 and 1) and write the sum (0 + 1 = 1) below the line. Finally, multiply the latest number below the line (1) by the root (1) and write the result (1 * 1 = 1) below the last coefficient (1). Add the numbers in the last column (1 and 1) and write the sum (1 + 1 = 2) below the line.

These repeated multiplications and additions form the core of the synthetic division algorithm. Each iteration brings us closer to the final result, systematically reducing the polynomial and revealing the quotient and remainder. The meticulous repetition of these steps ensures accuracy and efficiency in the division process.

  1. Interpreting the Result: The numbers below the horizontal line represent the coefficients of the quotient and the remainder. The last number (2) is the remainder. The other numbers (1, 1, 1) are the coefficients of the quotient, starting with a power one less than the dividend. Since our dividend was a cubic polynomial (x³), the quotient will be a quadratic polynomial. Therefore, the quotient is 1x² + 1x + 1, or simply x² + x + 1. The remainder is 2.

This final interpretation step is where we translate the numerical results of synthetic division back into polynomial form. By understanding the relationship between the numbers below the line and the coefficients of the quotient and remainder, we can effectively complete the division process and arrive at the solution. The remainder is a crucial piece of information, indicating whether the divisor divides the dividend evenly or not.

The Quotient and Remainder: Unveiling the Solution

From the synthetic division calculations, we have obtained the quotient and the remainder. The numbers below the line, excluding the last one, represent the coefficients of the quotient. In our example, these numbers are 1, 1, and 1. Since we started with a cubic polynomial (x³), the quotient will be a quadratic polynomial. Thus, the quotient is x² + x + 1.

The last number below the line represents the remainder. In our case, the remainder is 2. This means that when we divide (x³ + 1) by (x - 1), we get a quotient of x² + x + 1 and a remainder of 2. We can express this result as:

(x³ + 1) ÷ (x - 1) = x² + x + 1 + 2/(x - 1)

This equation represents the complete solution to our division problem. It shows how the dividend can be expressed in terms of the quotient, divisor, and remainder. The quotient (x² + x + 1) is the polynomial that results from the division, while the remainder (2) is the amount