Synthetic Division Step-by-Step Guide (x^3 - X^2 - 17x - 15) ÷ (x - 5)

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Introduction to Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a linear expression of the form x - a. It's an efficient alternative to long division, particularly when dealing with higher-degree polynomials. This method simplifies the division process by focusing on the coefficients of the polynomial and the root of the linear divisor. In this comprehensive guide, we will delve into the intricacies of synthetic division, providing a step-by-step solution to the problem (x³ - x² - 17x - 15) ÷ (x - 5). We will explore the underlying principles, demonstrate each step in detail, and highlight the advantages of using synthetic division for polynomial division. Understanding the mechanics of synthetic division not only enhances your ability to solve polynomial division problems but also lays a strong foundation for more advanced algebraic concepts.

Understanding Polynomial Division

Polynomial division is a fundamental operation in algebra that allows us to divide one polynomial by another. Similar to long division with numbers, polynomial division involves breaking down a complex polynomial into simpler terms. This process is crucial for various algebraic manipulations, such as factoring polynomials, finding roots, and simplifying rational expressions. The general form of polynomial division can be represented as P(x) ÷ D(x), where P(x) is the dividend (the polynomial being divided) and D(x) is the divisor (the polynomial we are dividing by). The result of this division is a quotient Q(x) and a remainder R(x), such that P(x) = D(x) * Q(x) + R(x). Mastering polynomial division techniques, including synthetic division, is essential for solving a wide range of algebraic problems and understanding the behavior of polynomial functions. In the following sections, we will focus specifically on synthetic division and its application to solving the given problem.

Setting Up Synthetic Division

Before we dive into the steps, it's crucial to understand how to set up a synthetic division problem correctly. The setup involves extracting the coefficients of the dividend polynomial and identifying the root of the divisor. In our problem, (x³ - x² - 17x - 15) ÷ (x - 5), the dividend is x³ - x² - 17x - 15, and the divisor is x - 5. First, we write down the coefficients of the dividend polynomial in a row: 1 (for x³), -1 (for x²), -17 (for x), and -15 (the constant term). It's important to include a coefficient of 0 for any missing terms in the polynomial. Next, we find the root of the divisor by setting x - 5 = 0 and solving for x, which gives us x = 5. This value, 5, is placed to the left of the coefficients. The setup is now complete, and we are ready to perform the synthetic division steps. This initial setup is the foundation for accurate and efficient polynomial division using this method.

Step-by-Step Solution using Synthetic Division

Now, let's walk through the synthetic division process step-by-step for the problem (x³ - x² - 17x - 15) ÷ (x - 5). This will provide a clear understanding of how the method works and how to arrive at the solution. The key steps involve bringing down, multiplying, adding, and repeating until we reach the end of the coefficients. Each step is crucial for obtaining the correct quotient and remainder.

  1. Bring Down: Begin by bringing down the first coefficient (1) below the line.
  2. Multiply: Multiply the brought-down coefficient (1) by the root of the divisor (5), resulting in 5.
  3. Add: Add the result (5) to the next coefficient (-1), giving us 4.
  4. Repeat: Multiply the sum (4) by the root (5), resulting in 20. Add this to the next coefficient (-17), giving us 3.
  5. Final Step: Multiply the sum (3) by the root (5), resulting in 15. Add this to the last coefficient (-15), giving us 0.

Interpreting the Results

After performing synthetic division, the numbers below the line represent the coefficients of the quotient and the remainder. In our case, the numbers below the line are 1, 4, 3, and 0. The last number, 0, is the remainder. The other numbers are the coefficients of the quotient polynomial. Since our original polynomial was of degree 3 (x³), and we divided by a linear term (x - 5), the quotient will be of degree 2. Therefore, the quotient is 1x² + 4x + 3, which can be written as x² + 4x + 3. The remainder is 0, indicating that (x - 5) divides the polynomial (x³ - x² - 17x - 15) evenly. This interpretation is a crucial step in understanding the outcome of synthetic division and its implications for polynomial factorization and root finding.

The Quotient

The quotient we obtained from the synthetic division is x² + 4x + 3. This result represents the polynomial that, when multiplied by the divisor (x - 5), gives us the original dividend (x³ - x² - 17x - 15). The quotient is a key component of polynomial division, providing valuable information about the relationship between the dividend and the divisor. In this case, the quotient is a quadratic polynomial, which can be further analyzed for its roots and factored form. Understanding the quotient is essential for various algebraic applications, including solving polynomial equations and simplifying rational expressions. The ability to accurately determine the quotient through synthetic division is a testament to the efficiency and power of this method.

Choosing the Correct Answer

Based on our synthetic division calculation, the quotient is x² + 4x + 3. Therefore, the correct answer from the given options is A. x² + 4x + 3. This step is crucial in any mathematical problem-solving process – ensuring that the final answer aligns with the calculations and interpretations made throughout the solution. By carefully performing synthetic division and correctly interpreting the results, we have confidently identified the quotient and selected the appropriate answer. This process reinforces the importance of accuracy and attention to detail in mathematical problem-solving.

Advantages of Synthetic Division

Synthetic division offers several advantages over long division when dividing polynomials by linear expressions. Its streamlined approach focuses on coefficients, making it faster and less prone to errors. The compact notation simplifies the process, especially for higher-degree polynomials. Synthetic division is also an excellent tool for evaluating polynomials at specific values, known as the Remainder Theorem. By using the root of the divisor, we can quickly find the remainder, which is the value of the polynomial at that root. This method is particularly useful in algebra and calculus for finding roots, factoring polynomials, and analyzing polynomial functions. The efficiency and versatility of synthetic division make it a valuable technique for anyone working with polynomials.

Common Mistakes to Avoid

While synthetic division is a powerful tool, it's essential to avoid common mistakes to ensure accurate results. One frequent error is forgetting to include a coefficient of 0 for missing terms in the polynomial. For example, if a polynomial is x⁴ - 2x² + 1, the coefficients should be 1, 0, -2, 0, and 1. Another mistake is using the wrong sign for the root of the divisor. If dividing by (x + 3), the root is -3, not 3. Careless arithmetic errors during the multiplication and addition steps can also lead to incorrect answers. To prevent these mistakes, double-check the setup, pay close attention to signs, and carefully perform each calculation. Practice and attention to detail are key to mastering synthetic division and avoiding these pitfalls.

Practice Problems

To solidify your understanding of synthetic division, practice with additional problems. Here are a few examples:

  1. (2x³ - 5x² + x + 2) ÷ (x - 2)
  2. (x⁴ + 3x³ - 2x² - 5x + 1) ÷ (x + 1)
  3. (3x³ + 8x² - 7x + 4) ÷ (x + 3)

Work through these problems step-by-step, applying the techniques we've discussed. Check your answers by multiplying the quotient by the divisor and adding the remainder; you should get the original dividend. Consistent practice will build your confidence and proficiency in using synthetic division to solve polynomial division problems.

Conclusion

In conclusion, synthetic division is a valuable and efficient method for dividing polynomials by linear expressions. By understanding the steps involved, from setting up the problem to interpreting the results, you can confidently solve polynomial division problems. The quotient we found for (x³ - x² - 17x - 15) ÷ (x - 5) is x² + 4x + 3, which corresponds to option A. Remember to avoid common mistakes and practice regularly to master this technique. Synthetic division is a fundamental skill in algebra, essential for various mathematical applications, including factoring polynomials, finding roots, and simplifying expressions. By mastering synthetic division, you'll enhance your problem-solving abilities and gain a deeper understanding of polynomial functions.