Synthetic Division: Divide (6x³ + 36x² + 5x + 30) By (x + 6)

Are you struggling with polynomial division? Synthetic division is a streamlined method for dividing a polynomial by a linear expression of the form x - k. It's much quicker and cleaner than long division, especially for higher-degree polynomials. In this comprehensive guide, we'll walk through the process step-by-step, using the example of dividing $6x^3 + 36x^2 + 5x + 30$ by $x + 6$. Let's dive in!

Understanding Synthetic Division

Before we jump into the example, let's grasp the core idea of synthetic division. It's a simplified way to divide polynomials when the divisor is a linear factor (like x + a or x - a). It focuses on the coefficients of the polynomial, making the process more efficient. Imagine you have a complex puzzle; synthetic division is like having a tool that helps you quickly fit the pieces together. We are going to solve this puzzle together!

The great thing about synthetic division is its efficiency. Instead of dealing with full polynomial expressions, you're working primarily with numbers. This reduces the chances of making mistakes and speeds up the entire process. In this article, we'll explore not just the how but also the why behind each step, ensuring you develop a solid understanding. Let's transform what might seem like a daunting task into a manageable and even enjoyable mathematical exercise.

Benefits of Using Synthetic Division

There are several benefits to using synthetic division:

• Efficiency: It's faster than long division, especially for higher-degree polynomials.
• Simplicity: It focuses on coefficients, reducing the complexity of the calculation.
• Reduced Errors: The streamlined process minimizes the chance of mistakes.
• Versatility: It can be used to find roots and factor polynomials.

Example: Dividing $6x^3 + 36x^2 + 5x + 30$ by $x + 6$

Let's break down the division of $6x^3 + 36x^2 + 5x + 30$ by $x + 6$ using synthetic division. This example will showcase every step, making it easier for you to follow along and apply the method to other problems. We’ll start with setting up the problem, then move through the calculations, and finally interpret the result. Prepare to see how synthetic division transforms a potentially complex problem into a straightforward process!

Step 1: Set Up the Synthetic Division

First, identify the coefficients of the polynomial and the root of the divisor.

• Polynomial: $6x^3 + 36x^2 + 5x + 30$
• Coefficients: 6, 36, 5, 30
• Divisor: $x + 6$
• Root: To find the root, set the divisor equal to zero: $x + 6 = 0$, so $x = -6$.

Now, set up the synthetic division table. Write the root (-6) outside to the left, and the coefficients (6, 36, 5, 30) in a row to the right. Leave a space below the coefficients and draw a horizontal line.

-6 | 6 36 5 30
   |__________________
   

Step 2: Perform the Division

1. Bring down the first coefficient (6) below the line.

   -6 | 6 36 5 30
      |__________________
      6
   

2. Multiply the root (-6) by the number you just brought down (6), and write the result (-36) under the next coefficient (36).

   -6 | 6 36 5 30
      |   -36
      |__________________
      6
   

3. Add the numbers in the second column (36 and -36) and write the sum (0) below the line.

   -6 | 6 36 5 30
      |   -36
      |__________________
      6 0
   

4. Multiply the root (-6) by the new number below the line (0), and write the result (0) under the next coefficient (5).

   -6 | 6 36 5 30
      |   -36 0
      |__________________
      6 0
   

5. Add the numbers in the third column (5 and 0) and write the sum (5) below the line.

   -6 | 6 36 5 30
      |   -36 0
      |__________________
      6 0 5
   

6. Multiply the root (-6) by the new number below the line (5), and write the result (-30) under the last coefficient (30).

   -6 | 6 36 5 30
      |   -36 0 -30
      |__________________
      6 0 5
   

7. Add the numbers in the last column (30 and -30) and write the sum (0) below the line. This final number is the remainder.

   -6 | 6 36 5 30
      |   -36 0 -30
      |__________________
      6 0 5 0
   

Step 3: Interpret the Result

The numbers below the line (6, 0, 5) are the coefficients of the quotient, and the last number (0) is the remainder. Since we started with a cubic polynomial (degree 3) and divided by a linear factor (degree 1), the quotient will be a quadratic polynomial (degree 2).

• Quotient: $6x^2 + 0x + 5 = 6x^2 + 5$
• Remainder: 0

Therefore, rac{6x^3 + 36x^2 + 5x + 30}{x + 6} = 6x^2 + 5.

More Examples and Practice Problems

To truly master synthetic division, it’s essential to practice with various examples. Try these problems on your own:

1. Divide $x^3 - 4x^2 + 2x - 3$ by $x - 2$.
2. Divide $2x^4 + 5x^3 - 2x + 8$ by $x + 3$.
3. Divide $3x^3 - 8x^2 + 10x - 4$ by $x - 1$.

Working through these examples will solidify your understanding and improve your speed and accuracy. Remember, the key is to follow the steps methodically and pay close attention to the signs.

Tips for Success

• Stay Organized: Keep your numbers aligned in columns to avoid mistakes.
• Watch the Signs: Pay close attention to positive and negative signs.
• Practice Regularly: The more you practice, the more comfortable you'll become with the process.

Common Mistakes to Avoid

Even with a clear method like synthetic division, it’s easy to slip up if you're not careful. Here are some typical errors to watch out for:

• Incorrectly Identifying the Root: Remember to solve x + k = 0 for x to find the correct root.
• Misaligning Columns: Keeping your numbers in neat columns is crucial for accurate calculations.
• Forgetting Placeholders: If a term is missing in the polynomial (e.g., no x term), use a 0 as a placeholder.
• Sign Errors: One of the most common mistakes is mixing up positive and negative signs during multiplication and addition.
• Misinterpreting the Result: Ensure you correctly interpret the coefficients as the terms of the quotient and identify the remainder.

By being aware of these common pitfalls, you can proactively avoid them and ensure your synthetic division calculations are accurate.

Conclusion

Synthetic division is a powerful tool for dividing polynomials, making complex problems much more manageable. By following the steps outlined above and practicing regularly, you can master this technique and tackle polynomial division with confidence. Remember, understanding the process is just as important as getting the right answer. So, take your time, be methodical, and enjoy the satisfaction of solving these problems with ease! We encourage you to continue exploring mathematical techniques and concepts. If you want to delve deeper into polynomial division and related topics, Khan Academy is an excellent resource.