Synthetic Division: Solve (2x³-14x²+x-4) / (x-7)

Have you ever struggled with dividing polynomials? It can seem daunting, but synthetic division offers a streamlined approach to tackle these problems. In this guide, we'll break down synthetic division, focusing on how to divide the polynomial $2x^3 - 14x^2 + x - 4$ by $x - 7$. We’ll cover each step in detail, ensuring you understand the process thoroughly. Whether you're a student grappling with algebra or just looking to brush up on your math skills, this article will provide a clear and comprehensive explanation.

Understanding Synthetic Division

Before diving into the specifics of our example, let's establish a solid understanding of what synthetic division is and why it's such a valuable tool. Synthetic division is a simplified method for dividing a polynomial by a linear divisor of the form $x - c$. It's a shorthand version of polynomial long division, offering a quicker and more efficient way to find the quotient and remainder. This technique is particularly useful when dealing with higher-degree polynomials, as it reduces the complexity of the division process. The key advantage of synthetic division lies in its focus on the coefficients of the polynomials, which streamlines the calculations and minimizes the chances of errors. By understanding the underlying principles, you can confidently apply synthetic division to a variety of polynomial division problems.

Benefits of Using Synthetic Division

Why choose synthetic division over traditional long division? The benefits are numerous. First and foremost, it's more efficient and faster. Synthetic division condenses the division process into a series of simple arithmetic operations, eliminating the need to write out the variables and exponents repeatedly. This not only saves time but also reduces the likelihood of making mistakes. Secondly, synthetic division is easier to learn and apply. The step-by-step process is straightforward, making it accessible to students and anyone who needs to divide polynomials. Finally, synthetic division is particularly useful when you need to find the remainder of a division, which can be crucial in various mathematical applications, such as the Remainder Theorem and the Factor Theorem. By mastering synthetic division, you'll add a powerful tool to your mathematical arsenal.

Setting Up the Synthetic Division for $(2x^3 - 14x^2 + x - 4) / (x - 7)$

Now, let's apply synthetic division to our specific problem: dividing $2x^3 - 14x^2 + x - 4$ by $x - 7$. The first step is to set up the synthetic division table correctly. To do this, we need to identify the coefficients of the polynomial and the value of $c$ from the divisor $x - c$. In our case, the polynomial is $2x^3 - 14x^2 + x - 4$, so the coefficients are 2, -14, 1, and -4. The divisor is $x - 7$, so $c$ is 7. Now, we write these values in the synthetic division table. We write the value of $c$ (which is 7) to the left, and the coefficients of the polynomial (2, -14, 1, and -4) to the right. Make sure to include a 0 for any missing terms in the polynomial (e.g., if there was no $x$ term, we would include a 0 for its coefficient). The setup is crucial for ensuring accurate calculations in the subsequent steps.

Detailed Steps for Setting Up the Table

To ensure clarity, let's break down the setup process into detailed steps. First, write down the coefficients of the polynomial in a row. In our example, these are 2, -14, 1, and -4. Make sure to include the signs (positive or negative) correctly. Next, identify the value of $c$ from the divisor $x - c$. In this case, the divisor is $x - 7$, so $c$ is 7. Write this value to the left of the coefficients, separated by a vertical line. Finally, draw a horizontal line below the coefficients, leaving space for the numbers that will be calculated during the division process. This table is now ready for the synthetic division steps. A well-organized table is key to avoiding errors and making the division process smoother.

Performing Synthetic Division: A Step-by-Step Walkthrough

With the table set up, we can now perform the synthetic division. This process involves a series of simple arithmetic operations: bringing down, multiplying, and adding. Let's go through each step for our example, $(2x^3 - 14x^2 + x - 4) / (x - 7)$. First, bring down the first coefficient (which is 2) below the horizontal line. Next, multiply this number by the value of $c$ (which is 7). So, 2 multiplied by 7 is 14. Write this result below the second coefficient (-14). Now, add the second coefficient and the result (i.e., -14 + 14), which gives us 0. Write this sum below the line. Repeat this process for the remaining coefficients. Multiply the new number (0) by $c$ (7), which gives 0. Write this below the third coefficient (1). Add 1 and 0, which gives 1. Write this sum below the line. Finally, multiply 1 by 7, which gives 7. Write this below the last coefficient (-4). Add -4 and 7, which gives 3. This final number is the remainder. By following these steps carefully, you can perform synthetic division accurately and efficiently.

Tips for Accurate Synthetic Division

To ensure accuracy when performing synthetic division, keep a few tips in mind. First, double-check your setup. Make sure you have written the coefficients correctly and identified the correct value of $c$. A mistake in the setup can lead to incorrect results. Second, pay close attention to the signs. A simple sign error can throw off the entire calculation. Third, work neatly and systematically. Keep your numbers aligned in columns to avoid confusion. If you're working on paper, use a ruler to draw straight lines. Fourth, check your work. After completing the synthetic division, you can verify your result by multiplying the quotient by the divisor and adding the remainder. This should give you the original polynomial. By following these tips, you can minimize errors and gain confidence in your synthetic division skills.

Interpreting the Results: Quotient and Remainder

After performing the synthetic division, we have a row of numbers below the horizontal line. These numbers represent the coefficients of the quotient and the remainder. In our example, the numbers are 2, 0, 1, and 3. The last number (3) is the remainder. The other numbers (2, 0, and 1) are the coefficients of the quotient. To determine the degree of the quotient, we subtract 1 from the degree of the original polynomial. In this case, the original polynomial is $2x^3 - 14x^2 + x - 4$, which has a degree of 3. So, the quotient will have a degree of 2. Therefore, the quotient is $2x^2 + 0x + 1$, which simplifies to $2x^2 + 1$. The remainder is 3. Now, we can express the result in the form q(x) + rac{r(x)}{b(x)}, where $q(x)$ is the quotient, $r(x)$ is the remainder, and $b(x)$ is the divisor. In our case, the result is (2x^2 + 1) + rac{3}{x - 7}. Understanding how to interpret these results is crucial for solving polynomial division problems.

Expressing the Result in the Required Form

To express the result in the form $q(x) + \frac{r(x)}{b(x)}$, we simply write the quotient, add the remainder divided by the divisor. In our example, the quotient $q(x)$ is $2x^2 + 1$, the remainder $r(x)$ is 3, and the divisor $b(x)$ is $x - 7$. So, the final result is $(2x^2 + 1) + \frac{3}{x - 7}$. This form clearly shows the quotient and the remainder, providing a complete solution to the division problem. It's important to present your answer in this format, especially when there is a remainder, as it accurately represents the relationship between the dividend, divisor, quotient, and remainder. This final step ensures that your solution is both mathematically correct and clearly communicated.

Conclusion: Mastering Synthetic Division

In conclusion, synthetic division is a powerful and efficient method for dividing polynomials, particularly when the divisor is a linear expression. By following the step-by-step process outlined in this guide, you can confidently tackle polynomial division problems. We've covered everything from setting up the synthetic division table to performing the calculations and interpreting the results. Remember, practice is key to mastering any mathematical technique. Work through various examples, and you'll soon find synthetic division becoming second nature. With this skill in your mathematical toolkit, you'll be well-equipped to handle more complex algebraic problems.

For further exploration of polynomial division and related concepts, visit Khan Academy's Polynomial Division Section.