Synthetic Division Explained Solve (x⁴ - 1) ÷ (x - 1)

Jul 13, 2025 by ADMIN 54 views

Mastering Synthetic Division A Comprehensive Guide to Solving (x^4-1) ÷ (x-1)

In the realm of polynomial division, synthetic division stands out as a streamlined and efficient technique, particularly when dividing by a linear factor. This article delves into the intricacies of synthetic division, providing a comprehensive guide to solving the problem (x⁴ - 1) ÷ (x - 1). We will break down the process step-by-step, ensuring a clear understanding of the method and its application. We will also discuss the underlying principles that make synthetic division such a powerful tool in algebra.

Understanding Synthetic Division

Synthetic division is a simplified method of polynomial division used when the divisor is a linear factor of the form (x - k), where k is a constant. It offers a more concise and organized approach compared to long division, especially for higher-degree polynomials. The key advantage of synthetic division lies in its ability to focus on the coefficients of the polynomials, eliminating the need to write out the variables and exponents repeatedly. This not only saves time but also reduces the chances of making errors. Before diving into the specific problem, let's outline the general steps involved in synthetic division:

Identify the coefficients of the dividend (the polynomial being divided) and write them in a row. Remember to include zeros for any missing terms. For example, if you have x⁴ + 2x² - 1, the coefficients would be 1, 0, 2, 0, and -1, representing the coefficients of x⁴, x³, x², x, and the constant term, respectively.
Determine the value of k from the divisor (x - k). In other words, find the value that makes the divisor equal to zero. For instance, if the divisor is (x - 3), then k = 3. If the divisor is (x + 2), then k = -2.
Set up the synthetic division table. Draw a horizontal line and a vertical line to create a space for the coefficients, the value of k, and the resulting quotient and remainder.
Bring down the first coefficient of the dividend below the horizontal line. This will be the first coefficient of the quotient.
Multiply the value of k by the number you just brought down and write the result below the next coefficient of the dividend.
Add the two numbers in the second column (the coefficient of the dividend and the result from the multiplication) and write the sum below the horizontal line. This will be the second coefficient of the quotient.
Repeat steps 5 and 6 for the remaining coefficients of the dividend. Each time, multiply the value of k by the last number written below the line, write the result below the next coefficient, and add the two numbers.
Interpret the result. The numbers below the horizontal line, excluding the last one, are the coefficients of the quotient. The last number is the remainder. The degree of the quotient is one less than the degree of the dividend.

Step-by-Step Solution for (x⁴ - 1) ÷ (x - 1)

Now, let's apply the synthetic division method to solve the given problem: (x⁴ - 1) ÷ (x - 1). This step-by-step solution will walk you through each stage of the process, clarifying any potential points of confusion and reinforcing your understanding of the technique.

1. Identify the Coefficients

The dividend is x⁴ - 1. We need to write down the coefficients, including zeros for any missing terms. The polynomial can be rewritten as x⁴ + 0x³ + 0x² + 0x - 1. Therefore, the coefficients are 1, 0, 0, 0, and -1. This step is crucial because it ensures that the place values are correctly aligned during the synthetic division process. Omitting a zero for a missing term can lead to an incorrect result.

2. Determine the Value of k

The divisor is (x - 1), so k = 1. This value is the root of the divisor and plays a central role in the synthetic division process. It's the number we'll use to multiply and add in the subsequent steps. Understanding how to find k is essential for applying synthetic division correctly in various problems.

3. Set Up the Synthetic Division Table

Draw a horizontal line and a vertical line. Write the value of k (which is 1) to the left of the vertical line. Then, write the coefficients (1, 0, 0, 0, -1) to the right of the vertical line, above the horizontal line. This setup provides a visual framework for organizing the calculations involved in synthetic division.

1 | 1 0 0 0 -1
  |______________________
  |

4. Bring Down the First Coefficient

Bring down the first coefficient (1) below the horizontal line. This is the first step in the iterative process of synthetic division. This number will be the leading coefficient of the quotient.

1 | 1 0 0 0 -1
  |______________________
  | 1

5. Multiply and Add (First Iteration)

Multiply the value of k (1) by the number you just brought down (1), which gives 1. Write this result below the next coefficient (0). Then, add the two numbers in the second column (0 and 1) and write the sum (1) below the horizontal line. This multiplication and addition step is the core of the synthetic division process, and it's repeated for each coefficient.

1 | 1 0 0 0 -1
  | 1__________________
  | 1 1

6. Multiply and Add (Second Iteration)

Multiply the value of k (1) by the last number written below the line (1), which gives 1. Write this result below the next coefficient (0). Then, add the two numbers in the third column (0 and 1) and write the sum (1) below the horizontal line. Notice how the process is repetitive, making synthetic division a systematic and efficient method.

1 | 1 0 0 0 -1
  | 1 1______________
  | 1 1 1

7. Multiply and Add (Third Iteration)

Multiply the value of k (1) by the last number written below the line (1), which gives 1. Write this result below the next coefficient (0). Then, add the two numbers in the fourth column (0 and 1) and write the sum (1) below the horizontal line. This step further demonstrates the iterative nature of synthetic division.

1 | 1 0 0 0 -1
  | 1 1 1__________
  | 1 1 1 1

8. Multiply and Add (Final Iteration)

Multiply the value of k (1) by the last number written below the line (1), which gives 1. Write this result below the last coefficient (-1). Then, add the two numbers in the last column (-1 and 1) and write the sum (0) below the horizontal line. This final step completes the synthetic division process, providing us with the coefficients of the quotient and the remainder.

1 | 1 0 0 0 -1
  | 1 1 1 1
  |______________________
  | 1 1 1 1 0

9. Interpret the Result

The numbers below the horizontal line, excluding the last one, are the coefficients of the quotient. In this case, they are 1, 1, 1, and 1. The last number (0) is the remainder. Since the original polynomial was of degree 4, the quotient will be of degree 3. Therefore, the quotient is 1x³ + 1x² + 1x + 1, which can be simplified to x³ + x² + x + 1. The remainder is 0, indicating that (x - 1) divides evenly into (x⁴ - 1).

The Quotient and the Correct Answer

Therefore, the quotient of (x⁴ - 1) ÷ (x - 1) is x³ + x² + x + 1. Comparing this result with the given options, we find that the correct answer is:

C. x³ + x² + x + 1

Benefits of Using Synthetic Division

Synthetic division offers several advantages over traditional long division, making it a valuable tool for polynomial manipulation. Its efficiency, simplicity, and reduced error potential make it a preferred method in many algebraic contexts. Let's explore some of the key benefits in detail:

Efficiency: Synthetic division is generally faster than long division, especially for higher-degree polynomials. The streamlined process focuses on the essential numerical calculations, minimizing the writing required and accelerating the division process. This efficiency is particularly noticeable when dealing with polynomials of degree 3 or higher.
Simplicity: The method is relatively straightforward and easy to learn. The step-by-step process is systematic and can be quickly mastered with practice. The focus on coefficients rather than the entire polynomial expression simplifies the calculations and reduces cognitive load.
Reduced Error Potential: By focusing on the coefficients and using a structured approach, synthetic division reduces the chances of making errors. The organization of the table helps keep track of the calculations, minimizing the risk of misplacing terms or making arithmetic mistakes. This is particularly crucial in complex polynomial divisions where errors can easily accumulate.
Applications in Root Finding: Synthetic division can be used to find the roots of polynomials. If the remainder is zero, it means that the divisor is a factor of the dividend, and the value of k is a root of the polynomial. This is a fundamental concept in algebra and is used extensively in solving polynomial equations.
Polynomial Factorization: Synthetic division can help factor polynomials. By finding one factor using synthetic division, you can reduce the degree of the polynomial and make it easier to find other factors. This iterative process of dividing and factoring is a powerful technique in polynomial algebra.
Remainder Theorem: Synthetic division is closely related to the Remainder Theorem, which states that when a polynomial f(x) is divided by (x - k), the remainder is f(k). Synthetic division provides a practical way to calculate this remainder, which has important implications in polynomial evaluation and analysis.

When to Use Synthetic Division

While synthetic division is a powerful tool, it's important to understand its limitations and when it's most appropriate to use. Synthetic division is specifically designed for dividing a polynomial by a linear factor of the form (x - k), where k is a constant. This is its primary domain of applicability, and attempting to use it with divisors of higher degree or more complex forms will lead to incorrect results. Understanding this limitation is crucial for using synthetic division effectively.

Here are some specific scenarios where synthetic division is the ideal choice:

Dividing by a Linear Factor: As mentioned earlier, synthetic division excels when the divisor is a linear expression of the form (x - k) or (ax - b). This is the most common application of synthetic division, and it provides a quick and efficient way to find the quotient and remainder.
Finding Roots of Polynomials: If you suspect that a particular value is a root of a polynomial, you can use synthetic division to test it. If the remainder is zero, then the value is indeed a root, and the quotient is the result of dividing the polynomial by (x - k). This is a direct application of the Factor Theorem, which states that (x - k) is a factor of f(x) if and only if f(k) = 0.
Factoring Polynomials: Once you find a root of a polynomial using synthetic division, you can use the quotient to factor the polynomial further. This iterative process of finding roots and factoring is a key technique in polynomial algebra.
Evaluating Polynomials (Remainder Theorem): According to the Remainder Theorem, the remainder when a polynomial f(x) is divided by (x - k) is equal to f(k). Synthetic division provides a convenient way to calculate this remainder, allowing you to evaluate the polynomial at a specific value without direct substitution.

However, it's equally important to recognize when synthetic division is not appropriate:

Dividing by Non-Linear Factors: Synthetic division cannot be used when the divisor is not a linear expression. For example, you cannot use synthetic division to divide a polynomial by (x² + 1) or (x³ - 2x + 3). In such cases, long division is the appropriate method.
Complex Divisors: Synthetic division is not suitable for divisors with complex coefficients or more complicated forms. The method relies on the simplicity of the linear factor, and more complex divisors require alternative techniques.

In summary, synthetic division is a powerful and efficient tool for dividing polynomials by linear factors. Its simplicity and speed make it a preferred method in many algebraic contexts. However, it's crucial to understand its limitations and use it only when appropriate. For divisions involving non-linear factors or more complex divisors, long division or other techniques should be employed.

Conclusion

In conclusion, synthetic division is a powerful and efficient technique for dividing polynomials by linear factors. By understanding the steps involved and practicing its application, you can master this valuable tool and solve polynomial division problems with ease. In the specific case of (x⁴ - 1) ÷ (x - 1), we have demonstrated how synthetic division leads to the quotient x³ + x² + x + 1, making option C the correct answer. This method not only simplifies the division process but also provides a deeper understanding of polynomial relationships and factorization.