Synthetic Division Remainder Explained Step-by-Step Solution

In the realm of polynomial algebra, synthetic division stands out as a streamlined technique for dividing a polynomial by a linear divisor. This method, a simplified version of polynomial long division, not only efficiently calculates the quotient but also elegantly reveals the remainder. Understanding remainders is crucial, as they hold significant implications in various mathematical contexts, from solving polynomial equations to grasping the factor and remainder theorems. This article delves into the concept of remainders within the framework of synthetic division, offering a comprehensive guide to interpreting and applying them.

At its core, synthetic division provides an efficient way to divide a polynomial, say, P(x), by a linear expression of the form x - c. The remainder theorem states that when P(x) is divided by x - c, the remainder is equal to P(c). This fundamental principle connects the remainder to the value of the polynomial at a specific point. The remainder obtained through synthetic division offers direct insight into whether the divisor is a factor of the polynomial. If the remainder is zero, it signifies that the divisor is indeed a factor, and c is a root of the polynomial equation P(x) = 0. Conversely, a non-zero remainder indicates that the divisor is not a factor, and the remainder represents the value of the polynomial at x = c. Synthetic division is a cornerstone technique in algebra, offering a swift and organized approach to polynomial division. Its efficiency stems from focusing solely on the coefficients of the polynomials, streamlining the division process. The remainder obtained not only completes the division process but also offers valuable insights into the polynomial's behavior, particularly concerning its factors and roots. In essence, mastering synthetic division and understanding the significance of remainders is indispensable for anyone delving into polynomial algebra and its applications.

Decoding the Synthetic Division Process

To effectively decode the synthetic division process, it's essential to understand the step-by-step methodology involved. Let's break down the process using the example provided: $1 floor 1 ext{ } 2 ext{ } 2 ext{ } -3 ext{ } 3$. This notation represents the division of a polynomial by x - 1. The coefficients of the polynomial are 1, 2, 2, -3, and 3, while 1 outside the division symbol indicates that we are dividing by x - 1. The first step involves bringing down the leading coefficient, which in this case is 1. This value becomes the first coefficient of the quotient. Next, we multiply this leading coefficient by the divisor, 1, and place the result under the second coefficient of the polynomial. This product, 1, is added to the second coefficient, 2, yielding a sum of 3. This sum becomes the next coefficient of the quotient. We then repeat the process: multiply the new quotient coefficient, 3, by the divisor, 1, and place the result under the next coefficient of the polynomial, which is 2. Adding these gives us 5. This iterative process continues for each coefficient. Multiply 5 by 1 and add it to -3, resulting in 2. Finally, multiply 2 by 1 and add it to 3, giving us 5. The final number obtained, 5, is the remainder. The other numbers (1, 3, 5, and 2) are the coefficients of the quotient polynomial, which is one degree lower than the original polynomial.

Understanding the arrangement and the iterative multiplication and addition steps is crucial for accurately performing synthetic division. Each number's position and role in the process contribute to the final result, which includes both the quotient and the remainder. The remainder, in particular, is a key indicator of whether the divisor is a factor of the polynomial, as dictated by the remainder theorem. In the given example, the non-zero remainder of 5 signifies that x - 1 is not a factor of the polynomial represented by the coefficients 1, 2, 2, -3, and 3. The quotient polynomial, represented by the coefficients 1, 3, 5, and 2, can be written as x³ + 3x² + 5x + 2. Thus, synthetic division not only simplifies polynomial division but also provides a clear pathway to finding the quotient and the remainder, offering a complete picture of the division process.

Identifying the Remainder: A Step-by-Step Solution

To identify the remainder in the synthetic division problem $1 floor 1 ext{ } 2 ext{ } 2 ext{ } -3 ext{ } 3$, we follow the synthetic division process meticulously. As described in the previous section, we bring down the leading coefficient (1), multiply it by the divisor (1), and add the result to the next coefficient. This process is repeated for each coefficient until we reach the end. The final number we obtain is the remainder. Let's go through the steps again:

1. Bring down the leading coefficient: 1
2. Multiply 1 by the divisor 1, and write the result (1) under the next coefficient (2).
3. Add 2 and 1 to get 3.
4. Multiply 3 by the divisor 1, and write the result (3) under the next coefficient (2).
5. Add 2 and 3 to get 5.
6. Multiply 5 by the divisor 1, and write the result (5) under the next coefficient (-3).
7. Add -3 and 5 to get 2.
8. Multiply 2 by the divisor 1, and write the result (2) under the next coefficient (3).
9. Add 3 and 2 to get 5.

The last number we obtained, 5, is the remainder. This remainder tells us that when the polynomial represented by the coefficients 1, 2, 2, -3, and 3 is divided by x - 1, the remainder is 5. This can be verified by expressing the original polynomial as x⁴ + 2x³ + 2x² - 3x + 3. According to the remainder theorem, the remainder when dividing by x - 1 is the value of the polynomial when x = 1. Substituting x = 1 into the polynomial gives us 1⁴ + 2(1)³ + 2(1)² - 3(1) + 3 = 1 + 2 + 2 - 3 + 3 = 5, confirming our result from the synthetic division.

Analyzing the Answer Choices

Now that we have determined the remainder to be 5, let's analyze the answer choices provided: A. 6, B. 4, C. 5, and D. 3. By carefully examining these options, we can clearly see that option C, which is 5, matches our calculated remainder. This reinforces the importance of the synthetic division process in accurately determining the remainder when dividing polynomials. The incorrect options serve as distractors, emphasizing the need for a thorough and methodical approach to solving synthetic division problems. Option A (6), B (4), and D (3) are all close in value to the correct answer, making it crucial to perform the synthetic division steps precisely to avoid errors. Understanding the significance of the remainder and its role in polynomial division is key to confidently selecting the correct answer. The remainder not only completes the division process but also provides crucial information about the relationship between the polynomial and the divisor. In this case, since the remainder is 5, we know that x - 1 is not a factor of the polynomial x⁴ + 2x³ + 2x² - 3x + 3. If the remainder were 0, x - 1 would be a factor, and 1 would be a root of the polynomial. Therefore, option C is the correct answer, as it accurately represents the remainder obtained through synthetic division and confirmed by the remainder theorem.

Conclusion: The Significance of Remainders

In conclusion, the remainder in the synthetic division problem $1 floor 1 ext{ } 2 ext{ } 2 ext{ } -3 ext{ } 3$ is 5, making option C the correct choice. The significance of remainders in synthetic division extends beyond just completing the division process. Remainders are fundamental to understanding the relationship between a polynomial and its divisors. The remainder theorem provides a direct link between the remainder and the value of the polynomial at a specific point, offering a powerful tool for evaluating polynomials and determining factors.

Synthetic division, with its streamlined approach, is an indispensable technique in polynomial algebra. It not only simplifies the division process but also yields valuable information about the polynomial's roots and factors. The remainder, in particular, serves as a critical indicator of whether a divisor is a factor of the polynomial. A zero remainder signifies that the divisor is indeed a factor, while a non-zero remainder indicates otherwise. This knowledge is crucial in various mathematical applications, including solving polynomial equations, simplifying rational expressions, and analyzing polynomial behavior. Furthermore, understanding remainders enriches one's comprehension of polynomial functions and their properties. It allows for a deeper exploration of algebraic concepts and enhances problem-solving skills in mathematics. The ability to accurately perform synthetic division and interpret remainders is a testament to mathematical proficiency and opens doors to more advanced topics in algebra and calculus. Therefore, mastering synthetic division and grasping the significance of remainders is not just about solving a specific type of problem but about building a solid foundation in polynomial algebra and its applications in various mathematical and scientific fields.

Final Answer: The final answer is (C).