Synthetic Division Of (x^3 + 1) ÷ (x - 1) Find The Quotient

Jul 16, 2025 by Admin 60 views

Use Synthetic Division to Solve (x^3 + 1) ÷ (x - 1) and Find the Quotient

In the realm of polynomial division, synthetic division emerges as a streamlined and efficient method, particularly adept at handling cases where the divisor is a linear expression. This article delves into the application of synthetic division to solve the problem (x^3 + 1) ÷ (x - 1), guiding you through the step-by-step process to determine the quotient. Understanding synthetic division empowers you to tackle polynomial division problems with confidence and accuracy.

Synthetic division serves as a simplified alternative to long division when dividing a polynomial by a linear expression of the form (x - c). Its elegance lies in its focus on coefficients, reducing the computational burden and making the process more efficient. The key to mastering synthetic division lies in understanding its setup and the iterative steps involved.

Before diving into the problem, let's outline the general steps of synthetic division:

Identify the coefficients: Extract the coefficients of the dividend polynomial and arrange them in descending order of powers of x. If any powers are missing, include a coefficient of 0 as a placeholder.
Determine the divisor's root: Set the divisor (x - c) equal to zero and solve for x. The value obtained, c, is the root used in the synthetic division process.
Set up the synthetic division table: Draw a horizontal line and a vertical line to create a table. Write the root (c) to the left of the vertical line and the coefficients of the dividend polynomial to the right of the vertical line.
Perform the iterative steps:
- Bring down the first coefficient below the horizontal line.
- Multiply the number just brought down by the root (c) and write the result below the next coefficient.
- Add the coefficient and the result obtained in the previous step and write the sum below the horizontal line.
- Repeat the multiplication and addition steps for all remaining coefficients.
Interpret the results: The numbers below the horizontal line represent the coefficients of the quotient polynomial, with the last number being the remainder. The degree of the quotient polynomial is one less than the degree of the dividend polynomial.

Now, let's apply synthetic division to solve the given problem: (x^3 + 1) ÷ (x - 1). We will meticulously follow each step to arrive at the quotient.

Identify the coefficients: The dividend polynomial is x^3 + 1. We can rewrite it as x^3 + 0x^2 + 0x + 1 to explicitly show all the powers of x. The coefficients are 1, 0, 0, and 1.
Determine the divisor's root: The divisor is (x - 1). Setting it equal to zero, we get x - 1 = 0, which gives us x = 1. So, the root is 1.
Set up the synthetic division table:
```
1 | 1  0  0  1
  |__________
```

Perform the iterative steps:

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

1 | 1 0 0 1 |__________ 1

*   Multiply the number just brought down (1) by the root (1) and write the result (1) below the next coefficient (0).

    ```
1 | 1  0  0  1
  |     1
  |__________
    1

Add the coefficient (0) and the result (1) and write the sum (1) below the horizontal line.

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

*   Multiply the last number below the line (1) by the root (1) and write the result (1) below the next coefficient (0).

    ```
1 | 1  0  0  1
  |     1  1
  |__________
    1  1

Add the coefficient (0) and the result (1) and write the sum (1) below the horizontal line.

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

*   Multiply the last number below the line (1) by the root (1) and write the result (1) below the last coefficient (1).

    ```
1 | 1  0  0  1
  |     1  1  1
  |__________
    1  1  1

Add the coefficient (1) and the result (1) and write the sum (2) below the horizontal line.

1 | 1 0 0 1 | 1 1 1 |__________ 1 1 1 2

Interpret the results: The numbers below the horizontal line are 1, 1, 1, and 2. These represent the coefficients of the quotient polynomial and the remainder. The quotient polynomial is x^2 + x + 1, and the remainder is 2.

Therefore, (x^3 + 1) ÷ (x - 1) = x^2 + x + 1 + 2/(x - 1).

From the synthetic division process, we have determined that the quotient is x^2 + x + 1 and the remainder is 2. This means that when we divide (x^3 + 1) by (x - 1), we obtain the polynomial x^2 + x + 1 with a remainder of 2. We can express this result as:

(x^3 + 1) ÷ (x - 1) = x^2 + x + 1 + 2/(x - 1)

Comparing this result with the given options, we can see that the correct answer is:

C. x^2 + x + 1 + 2/(x - 1)

It's important to understand why the other options are incorrect to solidify your understanding of synthetic division and polynomial division in general. Let's briefly examine each incorrect option:

A. x^2 + x + 1 + 2/(x + 1): This option has the correct quotient but an incorrect remainder term. The denominator in the remainder should match the original divisor (x - 1), not (x + 1).
B. x^2 - x + 1: This option provides an incorrect quotient. The signs of the terms in the quotient polynomial are not correct.
D. x^3 - x^2 + x: This option is completely incorrect as it does not represent the quotient obtained from the synthetic division process. The degree of this polynomial is also incorrect.

This article has provided a comprehensive walkthrough of using synthetic division to solve the problem (x^3 + 1) ÷ (x - 1). We meticulously followed each step of the synthetic division process, from identifying coefficients and the divisor's root to setting up the table and performing the iterative calculations. By correctly interpreting the results, we identified the quotient as x^2 + x + 1 and the remainder as 2, leading us to the final answer: x^2 + x + 1 + 2/(x - 1). Mastering synthetic division is a valuable asset in your mathematical toolkit, enabling you to efficiently solve polynomial division problems. Remember to practice consistently to solidify your understanding and build confidence in applying this powerful technique. Synthetic division is very useful for dividing polynomials and can help simplify complex mathematical problems. The result we found demonstrates how polynomial division works and offers a good example of a synthetic division problem solved step by step.

To further enhance your understanding and proficiency in synthetic division, consider these additional tips:

Practice Regularly: The key to mastering any mathematical technique is consistent practice. Work through a variety of synthetic division problems with different polynomials and divisors. This will help you internalize the steps and develop your problem-solving skills.
Pay Attention to Signs: Synthetic division involves careful manipulation of signs. A small error in sign can lead to an incorrect result. Double-check your calculations, especially when dealing with negative coefficients and roots. Careful attention to signs is a core aspect of math and contributes to more accurate solutions.
Handle Missing Terms: When the dividend polynomial has missing terms (e.g., no x^2 term), remember to include a coefficient of 0 as a placeholder. This ensures that the synthetic division process is performed correctly. Missing terms can drastically change the outcome, so using placeholders is necessary for accuracy.
Connect to Long Division: If you're struggling with synthetic division, try connecting it to long division. Understand how the steps in synthetic division correspond to the steps in long division. This can provide a deeper understanding of the underlying principles. Comparing synthetic and long division can help learners grasp polynomial division concepts better.
Verify Your Results: After performing synthetic division, you can verify your result by multiplying the quotient by the divisor and adding the remainder. The result should be equal to the original dividend polynomial. This step is vital for verifying results and ensuring no calculation errors occurred.
Use Online Resources: Numerous online resources, including videos and practice problems, can help you learn and practice synthetic division. Take advantage of these resources to supplement your learning. Online resources offer an extra layer of comprehension through varied explanations and practical demonstrations.

Polynomial division, including synthetic division, is not merely an abstract mathematical concept. It has practical applications in various fields, including:

Engineering: Polynomial division is used in control systems, signal processing, and circuit analysis. Engineers use these techniques to model and analyze systems described by polynomials.
Computer Graphics: Polynomial division is used in computer graphics for curve and surface design. Bezier curves and splines, which are used to create smooth curves and surfaces, are based on polynomial equations. Polynomials are therefore very useful in digital design and graphics.
Economics: Polynomial division can be used in economic modeling to analyze cost functions and revenue functions. These models help economists understand how costs and revenues change as a function of production levels.
Cryptography: Polynomial division plays a role in certain cryptographic algorithms, where polynomials are used to encode and decode information. Cryptography, by nature, utilizes complex mathematical constructs to ensure data security.

Understanding the real-world applications of polynomial division can motivate you to learn and master this technique. It demonstrates that mathematical concepts are not just theoretical exercises but powerful tools with broad applicability.

While synthetic division is a streamlined process, certain common mistakes can lead to incorrect results. Being aware of these pitfalls can help you avoid them.

Forgetting Placeholders: As mentioned earlier, forgetting to include placeholders (coefficients of 0) for missing terms in the dividend polynomial is a common mistake. This can significantly alter the outcome of the division. Always ensure that all powers of x, from the highest to the constant term, are represented with a coefficient, even if it's 0.
Incorrect Sign Manipulation: Synthetic division involves adding and multiplying numbers, and incorrect sign manipulation can lead to errors. Pay close attention to the signs of the coefficients and the root of the divisor. Careful sign tracking is a necessary aspect of mathematical precision.
Misinterpreting the Remainder: The last number in the bottom row of the synthetic division table represents the remainder. Misinterpreting this number can lead to an incorrect final answer. Always double-check that you've correctly identified the remainder.
Using the Wrong Root: Remember to use the root of the divisor (c in x - c) in the synthetic division process. Using the incorrect root will result in a wrong quotient and remainder. The root of the divisor is fundamental to initiating and completing the synthetic division properly.

By being mindful of these common mistakes and taking steps to avoid them, you can increase your accuracy and confidence in synthetic division.

In conclusion, this article has provided a detailed guide on how to use synthetic division to solve the problem (x^3 + 1) ÷ (x - 1) and identify the quotient. We have broken down the process into manageable steps, explained the rationale behind each step, and highlighted common mistakes to avoid. Synthetic division, as we have shown, is an efficient way to deal with polynomial division especially when the divisor is a simple linear factor.

Furthermore, we've expanded on the basic method by offering additional tips for mastering synthetic division, exploring its real-world applications, and providing insight into why the incorrect answer choices are wrong. Through consistent practice and application of these strategies, you can enhance your problem-solving skills and broaden your mathematical understanding. Polynomial division, including the use of synthetic division, has applications across various fields, from engineering to computer graphics, underscoring its practical importance. Mastery of this technique not only facilitates problem-solving but also offers a valuable skill set applicable in diverse contexts. The mastery of polynomial division through methods like synthetic division enhances problem-solving capabilities in multiple domains.

Remember, consistent practice and a clear understanding of the underlying principles are key to success in mathematics. So, keep practicing, keep exploring, and keep building your mathematical prowess!