Synthetic Division And Quotient Polynomial Calculation

Synthetic division is a streamlined method for dividing a polynomial by a linear divisor of the form x - c. It's a valuable tool in algebra, particularly when dealing with polynomial factorization and root finding. This article will walk you through the process of completing a synthetic division problem and identifying the quotient in polynomial form. We'll break down each step, ensuring a clear understanding of the technique. The provided problem presents a polynomial division using synthetic division, and our goal is to accurately complete the process and express the quotient as a polynomial. Before diving into the specific problem, let's first understand the fundamental steps involved in synthetic division. This understanding will make the process easier to follow and less prone to errors. Understanding the mechanics of synthetic division is crucial for successfully dividing polynomials. It's a technique that simplifies the long division process, especially when dealing with linear divisors. By mastering synthetic division, students can efficiently find the quotient and remainder, which are essential in various algebraic manipulations and problem-solving scenarios. In this detailed explanation, we'll cover not only the steps but also the underlying logic behind each operation, helping you grasp the concept more thoroughly. Mastering synthetic division is a significant step in understanding polynomial algebra. It enables you to perform polynomial division more efficiently and accurately, a skill that is essential in various mathematical contexts. Through this comprehensive guide, you'll gain the confidence and proficiency needed to tackle synthetic division problems effectively.

Problem

Complete the synthetic division problem below:

- 3 | 1  2  - 2  3

What is the quotient in polynomial form?

A. $x^2-x+1$ B. $x^2+x-3$ C. $x^2-x+3$ D. $x^2+x-1$

Steps to Solve Synthetic Division

To solve the given synthetic division problem, we will follow these fundamental steps:

1. Set up the Synthetic Division: Write the coefficients of the dividend polynomial (the polynomial being divided) in a row. Write the value of c (from the divisor x - c) to the left. In this case, the divisor is implicitly x + 3 (since we have -3 on the left), and the dividend is $x^3 + 2x^2 - 2x + 3$. The coefficients are 1, 2, -2, and 3. Setting up the synthetic division correctly is the foundation for accurate calculation. This step involves identifying the coefficients of the dividend polynomial and the constant term from the divisor. A proper setup ensures that the subsequent calculations align correctly, leading to the right quotient and remainder. Attention to detail in this initial stage is vital for avoiding errors. The setup not only arranges the numbers but also reflects the structure of the polynomial division process. It's a visual representation that makes the algebraic manipulation more accessible. Therefore, taking the time to set up the synthetic division accurately will significantly enhance your ability to solve the problem efficiently and correctly.
2. Bring Down the First Coefficient: Bring down the first coefficient of the dividend polynomial to the bottom row. This first coefficient will be part of the quotient. Bringing down the first coefficient is the initial step in the iterative process of synthetic division. This step is straightforward but crucial as it sets the stage for the subsequent multiplication and addition operations. The first coefficient essentially becomes the leading coefficient of the quotient polynomial. This process acts as the starting point for generating the terms of the quotient. It’s a fundamental action that transitions the problem from the setup phase to the calculation phase. This single action initiates the chain of operations that leads to the solution. Therefore, ensuring accuracy at this stage is paramount for the rest of the calculation.
3. Multiply and Add: Multiply the value on the left (c) by the number you just brought down, and write the result under the next coefficient. Add the two numbers in that column and write the sum in the bottom row. This step is the core of the synthetic division process, where multiplication and addition are alternately performed. Multiplying the value on the left by the number brought down and then adding the result to the next coefficient allows us to systematically reduce the polynomial's degree. This iterative process continues until all coefficients have been processed. The multiply-and-add step efficiently combines terms and handles the distribution of the division, making synthetic division a streamlined method. Accuracy in both the multiplication and addition steps is vital to prevent errors from propagating through the calculation. This step's repeated application allows for the gradual construction of the quotient and remainder, showcasing the power of the synthetic division method.
4. Repeat: Repeat the multiply and add steps for the remaining coefficients. Each repetition contributes to the determination of the coefficients of the quotient and the remainder. The repetitive nature of this step makes it crucial to understand the pattern and perform each operation accurately. This iterative process simplifies what would otherwise be a more complex long division procedure. By consistently applying the multiply-and-add steps, we systematically break down the dividend polynomial. This repeated process is a testament to the efficiency of synthetic division, allowing for a streamlined approach to polynomial division. Understanding this repetitive cycle is key to mastering synthetic division and confidently solving such problems.
5. Determine the Quotient and Remainder: The numbers in the bottom row (excluding the last one) are the coefficients of the quotient polynomial. The last number is the remainder. The final step in synthetic division involves interpreting the results obtained in the bottom row. The numbers, excluding the last one, form the coefficients of the quotient polynomial, while the last number represents the remainder of the division. Understanding how to translate these numbers into polynomial form is critical for arriving at the final answer. This interpretation step connects the numerical results of synthetic division to the algebraic representation of the quotient and remainder. Accurately identifying the quotient and remainder allows for a complete understanding of the polynomial division performed. This final step solidifies the understanding of the entire synthetic division process.

Performing the Calculation

Let's apply these steps to our problem:

- 3 | 1  2  - 2  3
    |________

1. Bring down the 1:

- 3 | 1  2  - 2  3
    |________
      1

2. Multiply -3 by 1 and write the result under 2, then add:

- 3 | 1  2  - 2  3
    |   - 3
    |________
      1 - 1

3. Multiply -3 by -1 and write the result under -2, then add:

- 3 | 1  2  - 2  3
    |   - 3  3
    |________
      1 - 1  1

4. Multiply -3 by 1 and write the result under 3, then add:

- 3 | 1  2  - 2  3
    |   - 3  3 - 3
    |________
      1 - 1  1  0

Identifying the Quotient

The numbers in the bottom row are 1, -1, and 1, with a remainder of 0. Since the original polynomial was of degree 3 ($x^3$), the quotient will be of degree 2. Thus, the quotient is $1x^2 - 1x + 1$, or $x^2 - x + 1$. The process of identifying the quotient from the bottom row involves understanding the degree reduction that occurs during synthetic division. The degree of the quotient polynomial is always one less than the degree of the dividend polynomial. Recognizing this relationship is crucial for accurately constructing the quotient polynomial from its coefficients. The coefficients in the bottom row, excluding the remainder, directly correspond to the coefficients of the quotient. This straightforward correspondence simplifies the task of writing the quotient in polynomial form. The final step of translating these coefficients into the correct polynomial expression completes the synthetic division process.

Final Answer

Therefore, the quotient in polynomial form is $x^2 - x + 1$.

The correct answer is A.

Conclusion

In conclusion, synthetic division provides an efficient method for dividing polynomials, especially when the divisor is a linear expression. By following the steps of setting up the division, bringing down the first coefficient, multiplying and adding, repeating the process, and identifying the quotient and remainder, one can effectively solve polynomial division problems. The given problem illustrates this process clearly, leading to the quotient $x^2 - x + 1$. Synthetic division is not just a computational tool; it's also a conceptual bridge connecting numerical operations with algebraic expressions. Mastering this technique enhances one's understanding of polynomial factorization, root finding, and other advanced algebraic concepts. The ability to perform synthetic division accurately and efficiently is a valuable asset in mathematics. It simplifies complex calculations and provides insights into the structure of polynomials and their relationships. Understanding the underlying principles and practicing the steps diligently will solidify your proficiency in synthetic division, empowering you to tackle a wide range of algebraic challenges with confidence. In summary, synthetic division is a powerful method that every student of algebra should master, and this detailed explanation provides the necessary tools and understanding to do so effectively.