Synthetic Division Form For (7x³ + X² - 4) ÷ (x - 5) Explained

In the realm of polynomial division, synthetic division stands out as a streamlined and efficient method, particularly when dividing by a linear factor. This article delves into the correct representation of the division problem ${(7x^3 + x^2 - 4) \div (x - 5)}$ in synthetic division form. We'll explore the underlying principles of synthetic division, its setup, and how to correctly translate a polynomial division problem into its synthetic division counterpart. Understanding this process is crucial for simplifying polynomial division and solving related algebraic problems. To effectively grasp the concept, we will dissect the components of the given polynomial and the divisor, highlighting the significance of each term and its placement in the synthetic division format. Synthetic division not only simplifies the calculation process but also provides a clear and organized way to handle polynomial divisions, making it an indispensable tool in algebra. By the end of this article, you will be able to confidently identify the correct synthetic division setup for any given polynomial division problem, ensuring accuracy and efficiency in your mathematical endeavors. The process involves extracting coefficients and using the root of the divisor to perform a series of multiplications and additions, ultimately leading to the quotient and remainder. Mastering synthetic division opens doors to solving complex polynomial equations and understanding the behavior of polynomial functions.

Deconstructing the Polynomial and Divisor

To accurately represent ${(7x^3 + x^2 - 4) \div (x - 5)}$ in synthetic division form, it’s essential to first deconstruct the polynomial and the divisor. The polynomial dividend is ${7x^3 + x^2 - 4}$, and the divisor is ${x - 5}$. When setting up synthetic division, we focus on the coefficients of the polynomial and the root of the divisor. The coefficients of the polynomial are the numerical values attached to each term, which in this case are 7 (for ${x^3}$), 1 (for ${x^2}$), 0 (for the missing ${x}$ term), and -4 (the constant term). It’s crucial to include a zero as a placeholder for any missing terms to maintain the correct order and degree of the polynomial. The divisor ${x - 5}$ provides the root, which is the value of ${x}$ that makes the divisor equal to zero. Solving ${x - 5 = 0}$ gives us ${x = 5}$, which is the value we will use in the synthetic division process. This root is placed outside the division symbol in the synthetic division setup. Understanding the role of coefficients and the root is paramount in correctly setting up and executing synthetic division. Neglecting a missing term or using the incorrect root will lead to an incorrect result. Therefore, a meticulous approach to identifying and extracting these components is necessary for successful polynomial division. This initial step lays the foundation for the subsequent calculations, ensuring that the synthetic division process accurately reflects the original polynomial division problem. The order of the coefficients is also crucial, as they represent the decreasing powers of ${x}$ in the polynomial.

Setting Up the Synthetic Division Format

The synthetic division format is a specific arrangement of numbers that allows for a streamlined division process. For the given problem, ${(7x^3 + x^2 - 4) \div (x - 5)}$, the setup involves writing the root of the divisor outside a division-like symbol and the coefficients of the dividend inside. As we determined earlier, the root of the divisor ${x - 5}$ is 5. The coefficients of the dividend ${7x^3 + x^2 - 4}$ are 7, 1, 0, and -4. It’s imperative to include the 0 for the missing ${x}$ term to maintain the proper place value. The synthetic division setup will look like this:

5 | 7 1 0 -4
  |____________

In this arrangement, the 5 is placed outside the division symbol, and the coefficients 7, 1, 0, and -4 are placed inside. A horizontal line is drawn below the coefficients to separate them from the results of the division process. This setup is the foundation for the synthetic division calculation. The first coefficient (7 in this case) is brought down below the line, and then the process of multiplication and addition begins. A correct setup is critical for accurate results. Any deviation from this format will lead to errors in the division process. The visual arrangement helps in organizing the calculations and minimizing mistakes. The line below the coefficients serves as a workspace for the intermediate results and the final quotient and remainder. This structured approach is what makes synthetic division an efficient and reliable method for polynomial division. Understanding the spatial arrangement and the significance of each number within the setup is key to mastering synthetic division.

Analyzing the Answer Choices

Given the problem ${(7x^3 + x^2 - 4) \div (x - 5)}$, let's analyze the answer choices to identify the correct synthetic division form. We've already established that the root of the divisor ${x - 5}$ is 5, and the coefficients of the dividend ${7x^3 + x^2 - 4}$ are 7, 1, 0, and -4. Now, we can evaluate each option:

• Option A: ${-5 \overline{) 7 \quad 1 \quad 0 \quad -4}}$ - This option incorrectly uses -5 as the root and is therefore incorrect.
• Option B: ${5 \overline{) 7 \quad 1 \quad 0 \quad -4}}$ - This option correctly uses 5 as the root and includes all the correct coefficients, making it the correct representation.
• Option C: ${-5 \overline{) 7 \quad 1 \quad -4}}$ - This option incorrectly uses -5 as the root and omits the 0 for the missing ${x}$ term, making it incorrect.
• Option D: ${5 \overline{) 7 \quad 1 \quad -4}}$ - This option correctly uses 5 as the root but omits the 0 for the missing ${x}$ term, making it incorrect.

Therefore, the correct answer is Option B, which accurately represents the synthetic division setup with the correct root and all the necessary coefficients, including the placeholder zero. Careful comparison of each option against the established criteria is essential for identifying the correct answer. The inclusion of the placeholder zero is a critical detail that distinguishes the correct answer from the incorrect ones. Understanding the significance of each element in the synthetic division setup allows for a systematic evaluation of the answer choices. This analytical approach ensures accuracy and reinforces the understanding of the synthetic division process.

The Correct Synthetic Division Form

Based on our analysis, the correct synthetic division form for ${(7x^3 + x^2 - 4) \div (x - 5)}$ is represented by Option B:

5 | 7 1 0 -4
  |____________

This representation accurately captures the essence of the division problem in the synthetic division format. The number 5, derived from the divisor ${x - 5}$, is placed outside the division symbol, representing the value at which we are evaluating the polynomial. The coefficients 7, 1, 0, and -4, extracted from the dividend ${7x^3 + x^2 - 4}$, are placed inside the division symbol, arranged in descending order of the powers of ${x}$. The inclusion of 0 for the missing ${x}$ term is crucial for maintaining the correct place value and ensuring an accurate result. This correct setup is the foundation for performing the synthetic division calculations. From here, we would bring down the first coefficient, multiply it by the root, add the result to the next coefficient, and repeat the process until we reach the end. The final row of numbers will then reveal the coefficients of the quotient and the remainder. Understanding and correctly setting up the synthetic division form is the first and most critical step in solving polynomial division problems efficiently. This foundational understanding allows for a smooth transition into the calculation phase, ensuring that the final result is both accurate and meaningful. The clarity and organization provided by the synthetic division format make it a powerful tool in polynomial algebra.

In conclusion, mastering the synthetic division form is a crucial skill in polynomial algebra. For the given problem, ${(7x^3 + x^2 - 4) \div (x - 5)}$, the correct representation in synthetic division form is Option B:

5 | 7 1 0 -4
  |____________

This setup accurately includes the root of the divisor (5) and all the coefficients of the dividend (7, 1, 0, -4), ensuring that the synthetic division process can be executed correctly. The inclusion of the placeholder zero for the missing ${x}$ term highlights the importance of attention to detail in mathematical operations. Understanding the underlying principles of synthetic division, including the roles of the divisor's root and the dividend's coefficients, is essential for success in polynomial division. By correctly setting up the synthetic division, we pave the way for efficient and accurate calculations. This method not only simplifies the division process but also provides a structured approach to solving complex algebraic problems. As we've seen, a careful analysis of the polynomial and divisor, combined with a thorough understanding of the synthetic division format, leads to the correct representation. This mastery empowers us to tackle more advanced polynomial manipulations and applications. The ability to confidently set up and execute synthetic division is a valuable asset in the study of algebra and beyond.