Synthetic Division Solving (4x^3 - 3x^2 + 5x + 6) ÷ (x + 6)

Jul 14, 2025 by ADMIN 60 views

Synthetic Division to Solve Polynomial Division Problems

When faced with polynomial division problems, particularly those involving a linear divisor, synthetic division offers a streamlined and efficient approach. This method simplifies the division process, making it easier to determine the quotient and remainder. In this article, we will delve into the application of synthetic division to solve the problem $(4x^3 - 3x^2 + 5x + 6) \\div (x + 6)$ , and thoroughly explain each step involved in finding the quotient.

Understanding Synthetic Division

Synthetic division is a shorthand method of dividing a polynomial by a linear divisor of the form $(x - k)$ . It leverages the coefficients of the polynomial and the value of $k$ to systematically reduce the polynomial's degree. Unlike long division, which can be cumbersome and prone to errors, synthetic division provides a concise and organized way to perform the division. This method is particularly useful in algebra and calculus, where polynomial division is a common operation.

Key benefits of using synthetic division include:

Efficiency: Synthetic division is generally faster and less prone to errors than long division, especially for linear divisors.
Organization: The method is structured and easy to follow, reducing the chances of making mistakes.
Versatility: Synthetic division can be used to find the roots of polynomials and factor them.

Step-by-Step Solution

To solve the given problem, $(4x^3 - 3x^2 + 5x + 6) \\div (x + 6)$ , we will follow these steps:

Step 1: Identify the Divisor and Dividend

The dividend is the polynomial being divided, which in this case is $4x^3 - 3x^2 + 5x + 6$ . The divisor is the polynomial we are dividing by, which is $x + 6$ . To use synthetic division, we need to express the divisor in the form $(x - k)$ . In this case, $x + 6$ can be rewritten as $x - (-6)$ , so $k = -6$ . Understanding the divisor and dividend is a foundational step in the synthetic division process.

Step 2: Set Up the Synthetic Division Table

Create a table with the coefficients of the dividend and the value of $k$ . Write the coefficients of the dividend in a row, ensuring that they are in descending order of powers of $x$ . If any power of $x$ is missing, include a coefficient of 0 as a placeholder. In our case, the coefficients are 4, -3, 5, and 6. Write the value of $k$ (which is -6) to the left of the coefficients. Setting up the table correctly is crucial for the accuracy of the subsequent steps. Here’s how the table should look:

-6 | 4 -3 5 6
   |________________________
   |

Step 3: Perform the Synthetic Division

Bring down the first coefficient (4) to the bottom row.
```
-6 | 4 -3 5 6
   |________________________
   | 4
```
Multiply the value of $k$ (-6) by the number in the bottom row (4) and write the result (-24) under the next coefficient (-3).
```
-6 | 4 -3 5 6
   | -24
   |________________________
   | 4
```
Add the numbers in the second column (-3 and -24) and write the sum (-27) in the bottom row.
```
-6 | 4 -3 5 6
   | -24
   |________________________
   | 4 -27
```
Multiply the value of $k$ (-6) by the new number in the bottom row (-27) and write the result (162) under the next coefficient (5).
```
-6 | 4 -3 5 6
   | -24 162
   |________________________
   | 4 -27
```
Add the numbers in the third column (5 and 162) and write the sum (167) in the bottom row.
```
-6 | 4 -3 5 6
   | -24 162
   |________________________
   | 4 -27 167
```
Multiply the value of $k$ (-6) by the new number in the bottom row (167) and write the result (-1002) under the last coefficient (6).
```
-6 | 4 -3 5 6
   | -24 162 -1002
   |________________________
   | 4 -27 167
```
Add the numbers in the last column (6 and -1002) and write the sum (-996) in the bottom row. This is the remainder.
```
-6 | 4 -3 5 6
   | -24 162 -1002
   |________________________
   | 4 -27 167 -996
```

Step 4: Interpret the Results

The numbers in the bottom row, excluding the last number, are the coefficients of the quotient. The last number is the remainder. In this case, the coefficients of the quotient are 4, -27, and 167. The remainder is -996. Interpreting the results correctly is essential for expressing the final answer.

Step 5: Write the Quotient and Remainder

The quotient is a polynomial with a degree one less than the dividend. Since the dividend was a cubic polynomial (degree 3), the quotient will be a quadratic polynomial (degree 2). Using the coefficients from the bottom row, the quotient is $4x^2 - 27x + 167$ . The remainder is -996. Therefore, the result of the division can be expressed as:

4x^2 - 27x + 167 - \frac{996}{x + 6}

Identifying the Correct Answer

Based on our calculations, the quotient is $4x^2 - 27x + 167$ and the remainder is -996. This means the correct answer is:

D. $4x^2 - 27x + 167 - \frac{996}{x + 6}$

Common Mistakes to Avoid

When performing synthetic division, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them:

Forgetting to include a 0 for missing terms: Ensure that all powers of $x$ are accounted for in the dividend. If a term is missing, include a 0 as its coefficient. For example, if dividing $x^3 + 1$ by $x - 1$ , the dividend should be considered as $x^3 + 0x^2 + 0x + 1$ .
Incorrectly identifying the value of k: Remember that the divisor must be in the form $(x - k)$ . If the divisor is $(x + k)$ , then the value of $k$ is negative. For instance, for a divisor of $x + 3$ , $k$ would be -3.
Making arithmetic errors: Synthetic division involves multiple steps of multiplication and addition. A simple arithmetic error can propagate through the process and lead to a wrong answer. Double-check each calculation to ensure accuracy.
Misinterpreting the result: The numbers in the bottom row represent the coefficients of the quotient and the remainder. Ensure you assign the correct powers of $x$ to the coefficients and express the remainder as a fraction over the divisor.

Practice Problems

To solidify your understanding of synthetic division, here are some practice problems:

Divide $(2x^3 - 5x^2 + 3x + 4) \\div (x - 2)$ .
Divide $(x^4 + 3x^3 - 2x^2 + 5x - 1) \\div (x + 1)$ .
Divide $(3x^3 + 8x^2 - 7x + 2) \\div (x + 3)$ .

Working through these problems will help you become more comfortable with the synthetic division process and improve your accuracy.

Conclusion

Synthetic division is a powerful tool for simplifying polynomial division, particularly when dealing with linear divisors. By following the steps outlined in this article, you can efficiently and accurately determine the quotient and remainder. Remember to pay close attention to the setup, perform the calculations carefully, and interpret the results correctly. With practice, you can master this technique and confidently solve polynomial division problems.

By understanding and applying synthetic division effectively, you can tackle complex polynomial division problems with ease and precision. This method not only saves time but also reduces the likelihood of errors, making it an invaluable tool in various mathematical contexts.