Is Synthetic Division Suitable For (4x² - 3x + 6) ÷ (x + 2)?

Is the division problem extbf{a candidate for the synthetic division process}? This question arises when we encounter polynomial division, specifically when dividing by a binomial. Synthetic division is a streamlined method for dividing polynomials, but it's not universally applicable. Let's delve into the specifics of synthetic division and determine its suitability for the given problem: $(4x^2 - 3x + 6) ÷ (x + 2)$. This article aims to provide a comprehensive understanding of synthetic division, its requirements, and its application to the given polynomial division problem. We will explore the conditions under which synthetic division is appropriate and demonstrate the process step-by-step to clarify its mechanics and limitations.

Understanding Synthetic Division

Synthetic division, a streamlined alternative to long division, is particularly useful for dividing polynomials by linear expressions. Synthetic division simplifies the division process, making it quicker and less prone to errors, especially when dealing with higher-degree polynomials. However, its applicability is limited to divisors of a specific form. The key advantage of synthetic division lies in its efficiency. It focuses on the coefficients of the polynomials, reducing the complexity of the calculation. This method eliminates the need to write out the variables and exponents repeatedly, which can be cumbersome in long division. By focusing solely on the numerical coefficients, synthetic division reduces the risk of making algebraic errors, such as misplacing terms or incorrectly applying the distributive property. It is particularly beneficial when dealing with polynomials of higher degrees, where the long division process can become quite lengthy and intricate. Moreover, synthetic division provides a clear and organized way to track the intermediate steps of the division, making it easier to identify and correct any mistakes. The process involves setting up a simple table-like structure where the coefficients of the dividend and the root of the divisor are arranged. Then, through a series of multiplications and additions, the quotient and the remainder are obtained. This structured approach not only enhances accuracy but also aids in understanding the relationship between the dividend, divisor, quotient, and remainder. This method is generally preferred for its speed and simplicity, making it an indispensable tool for polynomial division when applicable. Synthetic division is a valuable tool in algebra, especially when dealing with polynomial equations and their roots.

Requirements for Synthetic Division

Before applying synthetic division, it's crucial to understand its limitations. extbf{Synthetic division} is specifically designed for dividing a polynomial by a binomial of the form $(x - c)$, where 'c' is a constant. This restriction is essential because the method relies on manipulating the coefficients of the polynomial and the constant term of the divisor. The divisor must be a linear expression with a leading coefficient of 1. If the divisor is not in this form, synthetic division cannot be directly applied. For instance, if the divisor is $(2x + 3)$, it needs to be manipulated into the required form before synthetic division can be used. This often involves dividing both the divisor and the dividend by the leading coefficient of the divisor. However, it's important to note that this manipulation can affect the quotient and remainder, requiring further adjustments to obtain the correct result. Another key requirement is that the dividend must be a polynomial. The terms of the polynomial must be arranged in descending order of their exponents, and any missing terms (e.g., if there's no $x$ term in a cubic polynomial) must be represented with a coefficient of 0. This ensures that the coefficients are aligned correctly during the synthetic division process. Understanding these requirements is crucial for determining whether synthetic division is the appropriate method for a given division problem. Attempting to use synthetic division with a divisor that doesn't meet the required form or with a non-polynomial dividend will lead to incorrect results. Therefore, a careful assessment of the problem is necessary before proceeding with synthetic division. This includes checking the form of the divisor, ensuring the dividend is a polynomial, and accounting for any missing terms in the dividend.

Analyzing the Given Problem: (4x² - 3x + 6) ÷ (x + 2)

Now, let's analyze the given problem: $(4x^2 - 3x + 6) ÷ (x + 2)$. The dividend is the polynomial $(4x^2 - 3x + 6)$, and the divisor is $(x + 2)$. To determine if synthetic division is suitable, we need to check if the divisor is in the form $(x - c)$. In this case, the divisor $(x + 2)$ can be rewritten as $(x - (-2))$, where $c = -2$. Since the divisor fits the required form, extbf{synthetic division} is indeed a viable method for solving this problem. The polynomial $4x^2 - 3x + 6$ is a quadratic polynomial, and the divisor $x + 2$ is a linear binomial. Synthetic division is particularly efficient for dividing polynomials by linear binomials of this form. The process involves setting up the coefficients of the dividend and the value of $c$ from the divisor in a specific format, which allows for a streamlined calculation of the quotient and remainder. This method is often preferred over long division because it reduces the complexity of the calculations and minimizes the chances of making errors. The coefficients of the dividend are 4, -3, and 6, corresponding to the terms $4x^2$, $-3x$, and the constant term, respectively. The value of $c$ is -2, which is obtained by setting the divisor $x + 2$ equal to zero and solving for $x$. With these values, we can set up the synthetic division tableau and proceed with the calculations. The ability to recognize when synthetic division is applicable and to correctly identify the values needed to set up the process is a crucial skill in algebra. This understanding allows for efficient and accurate solutions to polynomial division problems.

Performing Synthetic Division: Step-by-Step

To perform synthetic division for $(4x^2 - 3x + 6) ÷ (x + 2)$, we'll follow these steps:

1. Identify the coefficients of the dividend and the value of 'c'. The coefficients of the dividend $(4x^2 - 3x + 6)$ are 4, -3, and 6. The divisor is $(x + 2)$, so $c = -2$. This step is crucial because it sets the foundation for the entire synthetic division process. The coefficients represent the numerical values associated with each term of the polynomial, and the value of 'c' is derived from the divisor. The correct identification of these values ensures the accuracy of the subsequent calculations. Misidentifying the coefficients or the value of 'c' can lead to an incorrect quotient and remainder. Therefore, it's essential to double-check these values before proceeding with the next steps. This careful preparation is a hallmark of effective problem-solving in algebra and helps to avoid common errors in synthetic division. The coefficients are arranged in the order of the descending powers of $x$, and any missing terms are represented with a coefficient of 0. The value of 'c' is the root of the divisor, which is obtained by setting the divisor equal to zero and solving for $x$.
2. Set up the synthetic division tableau. Write 'c' (-2) to the left. Then, write the coefficients of the dividend (4, -3, 6) to the right. Draw a horizontal line below the coefficients. Setting up the tableau correctly is essential for organizing the synthetic division process. The tableau provides a visual framework for the calculations, ensuring that the numbers are aligned properly and the steps are followed in the correct order. The value of 'c' is placed to the left as it is the value by which the intermediate results will be multiplied. The coefficients of the dividend are written in a row, representing the polynomial terms in descending order of their exponents. The horizontal line separates the coefficients of the dividend from the intermediate results and the coefficients of the quotient and the remainder. This setup allows for a systematic calculation, making it easier to track the steps and minimize errors. The tableau is a fundamental tool in synthetic division, and mastering its setup is crucial for efficient and accurate problem-solving.
3. Bring down the first coefficient. Bring down the first coefficient (4) below the horizontal line. This is the first step in the iterative process of synthetic division. The first coefficient, which represents the leading coefficient of the dividend, is brought down directly as it is the starting point for the calculations. This step is straightforward but essential as it initiates the chain of multiplications and additions that will lead to the quotient and remainder. The brought-down coefficient becomes the leading coefficient of the quotient polynomial. This step highlights the connection between the coefficients of the dividend and the coefficients of the quotient. Understanding this connection is crucial for interpreting the results of synthetic division and expressing the quotient and remainder correctly. The act of bringing down the first coefficient sets the stage for the subsequent steps, where the value of 'c' is used to modify the coefficients and ultimately determine the result of the division.
4. Multiply and add. Multiply 'c' (-2) by the number you brought down (4), which gives -8. Write -8 below the next coefficient (-3). Add -3 and -8, which gives -11. This step is the core of the synthetic division process. It involves a repeated cycle of multiplication and addition that systematically reduces the degree of the polynomial and determines the coefficients of the quotient and the remainder. The value of 'c' is multiplied by the previously brought-down number, and the result is written below the next coefficient in the dividend. This multiplication step effectively accounts for the shift in the terms during polynomial division. The addition step then combines the result of the multiplication with the corresponding coefficient of the dividend, which effectively subtracts the appropriate multiple of the divisor from the dividend. This process is repeated for each coefficient, gradually revealing the quotient and remainder. The accuracy of this step is paramount, as any error in multiplication or addition will propagate through the rest of the synthetic division, leading to an incorrect answer. Therefore, careful attention and precision are required during this crucial phase.
5. Repeat the multiply and add process. Multiply 'c' (-2) by -11, which gives 22. Write 22 below the last coefficient (6). Add 6 and 22, which gives 28. The repetition of the multiply and add process is what makes synthetic division such an efficient method for polynomial division. This iterative cycle ensures that each coefficient of the dividend is properly accounted for and that the effects of the divisor are systematically incorporated. By repeatedly multiplying 'c' by the latest result and adding it to the next coefficient, the process effectively performs the long division algorithm in a condensed and streamlined manner. This step highlights the recursive nature of synthetic division, where each calculation builds upon the previous one. The result of each addition becomes the input for the next multiplication, creating a chain reaction that ultimately reveals the quotient and remainder. The final addition step yields the remainder, while the other results form the coefficients of the quotient polynomial. The ability to perform this step accurately and efficiently is a key skill in mastering synthetic division.
6. Interpret the results. The numbers below the horizontal line (excluding the last one) are the coefficients of the quotient. The last number is the remainder. In this case, the quotient is $4x - 11$, and the remainder is 28. Interpreting the results of synthetic division correctly is crucial for expressing the final answer in the appropriate form. The coefficients of the quotient are read from left to right, with each coefficient corresponding to a term of decreasing degree. The degree of the quotient polynomial is one less than the degree of the dividend, reflecting the division by a linear divisor. The last number in the row is the remainder, which represents the amount left over after the division. This remainder can be expressed as a fraction with the divisor as the denominator, or it can be left as a separate term. Understanding the meaning of the quotient and remainder is essential for applying the results of synthetic division to solve various algebraic problems, such as factoring polynomials, finding roots, and simplifying rational expressions. The quotient and remainder provide a complete picture of the division, showing how the dividend can be expressed in terms of the divisor.

Conclusion

In conclusion, the division problem $(4x^2 - 3x + 6) ÷ (x + 2)$ is indeed a candidate for the synthetic division process. This is because the divisor $(x + 2)$ is a binomial of the form $(x - c)$, where $c = -2$. Synthetic division provides an efficient method for solving this problem, as demonstrated by the step-by-step process outlined above. The result of the division is a quotient of $(4x - 11)$ and a remainder of 28. This confirms that synthetic division is a valuable tool for polynomial division when the divisor is a linear binomial of the form $(x - c)$. Understanding the requirements and limitations of synthetic division is crucial for applying it correctly and efficiently. While it is not a universal method for all polynomial division problems, it is a powerful technique for specific cases. The ability to recognize when synthetic division is appropriate and to execute the process accurately is a valuable skill in algebra and beyond. The results obtained from synthetic division can be used for various applications, such as simplifying rational expressions, finding roots of polynomials, and solving polynomial equations.