Synthetic Division Suitability For (2x^7-6) ÷ (x^5+4)

Is the division problem ${(2x^7 - 6) \div (x^5 + 4)}$ a candidate for the synthetic division process? This is a crucial question when dealing with polynomial division in mathematics. To answer this, we need to understand the fundamental principles behind synthetic division and its applicability. This article will delve deep into the requirements for using synthetic division, analyze the given problem, and provide a clear, concise answer. We will also explore alternative methods for polynomial division and highlight the significance of choosing the right approach to solve such problems efficiently.

Understanding Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a linear binomial. It simplifies the long division process, making it quicker and less prone to errors. However, its applicability is limited to cases where the divisor is a linear expression of the form ${x - c}$, where ${c}$ is a constant. Before we dive into whether our specific problem fits the bill, let's break down the key characteristics and requirements for using synthetic division.

Key Requirements for Synthetic Division

To effectively employ synthetic division, several conditions must be met. Understanding these prerequisites is essential for determining whether synthetic division is the appropriate method for a given problem.

1. Divisor Must Be Linear: The divisor must be a linear binomial, meaning it should be in the form ${x - c}$, where ${c}$ is a constant. For instance, ${x - 2}$, ${x + 3}$, and ${x - 1/2}$ are all linear binomials suitable for synthetic division. Expressions like ${x^2 + 1}$ or ${x^3 - 8}$ are not linear and, therefore, not candidates for synthetic division.
2. Polynomial Must Be in Standard Form: The dividend, which is the polynomial being divided, should be written in standard form. This means the terms should be arranged in descending order of their exponents. For example, the polynomial ${3x^4 - 2x^2 + 5x - 1}$ is in standard form, while ${-2x^2 + 3x^4 + 5x - 1}$ is not. Ensuring the polynomial is in standard form helps in correctly setting up the synthetic division process.
3. Include Zero Coefficients: If any terms are missing in the polynomial (i.e., if a particular power of ${x}$ is absent), a zero coefficient must be included as a placeholder. For example, if we have the polynomial ${x^4 - 1}$, it should be treated as ${x^4 + 0x^3 + 0x^2 + 0x - 1}$ for synthetic division. These zero coefficients maintain the correct place values and ensure the accuracy of the division process.

The Mechanics of Synthetic Division

The process of synthetic division involves a series of steps that efficiently compute the quotient and remainder when dividing a polynomial by a linear binomial. Here’s a detailed breakdown of the procedure:

1. Write Down the Coefficients: Begin by writing down the coefficients of the polynomial in a row. Ensure that the polynomial is in standard form and that zero coefficients are included for any missing terms. For example, if dividing ${x^3 - 2x + 1}$ by ${x - 3}$, the coefficients would be 1, 0, -2, and 1.
2. Identify the Constant Term: From the divisor ${x - c}$, identify the constant term ${c}$. In our example of dividing by ${x - 3}$, the constant term ${c}$ is 3. This value will be used in the synthetic division process.
3. Set Up the Synthetic Division Table: Draw a horizontal line and a vertical line to create a table. Write the constant term ${c}$ to the left of the vertical line and the coefficients of the polynomial to the right of the vertical line, above the horizontal line.
4. Bring Down the First Coefficient: Bring down the first coefficient of the polynomial below the horizontal line. This first coefficient will be the leading coefficient of the quotient.
5. Multiply and Add: Multiply the constant term ${c}$ by the number you just brought down, and write the result below the next coefficient. Add the two numbers in that column and write the sum below the horizontal line. Repeat this process for each subsequent coefficient.
6. Interpret the Results: The numbers below the horizontal line represent the coefficients of the quotient and the remainder. The last number is the remainder, and the other numbers are the coefficients of the quotient, starting with one degree less than the original polynomial.

Example of Synthetic Division

Let’s illustrate synthetic division with an example. Suppose we want to divide ${2x^3 - 5x^2 + 3x - 10}$ by ${x - 2}$. Here’s how we would perform synthetic division:

1. Coefficients: The coefficients of the polynomial are 2, -5, 3, and -10.
2. Constant Term: The constant term ${c}$ from the divisor ${x - 2}$ is 2.
3. Set Up:

   2 | 2 -5 3 -10
     |__________
   

4. Bring Down: Bring down the first coefficient, 2.

   2 | 2 -5 3 -10
     | 2________
   

5. Multiply and Add:
   • Multiply 2 (the constant term) by 2 (the brought-down coefficient) to get 4. Write 4 below -5 and add them: -5 + 4 = -1.

     2 | 2 -5 3 -10
       | 4
       |__________
         2 -1
     

   • Multiply 2 by -1 to get -2. Write -2 below 3 and add them: 3 + (-2) = 1.

     2 | 2 -5 3 -10
       | 4 -2
       |__________
         2 -1 1
     

   • Multiply 2 by 1 to get 2. Write 2 below -10 and add them: -10 + 2 = -8.

     2 | 2 -5 3 -10
       | 4 -2 2
       |__________
         2 -1 1 -8
     

6. Interpret: The numbers 2, -1, and 1 are the coefficients of the quotient, and -8 is the remainder. The quotient is ${2x^2 - x + 1}$, and the remainder is -8.

Thus, ${(2x^3 - 5x^2 + 3x - 10) \div (x - 2) = 2x^2 - x + 1 - \frac{8}{x - 2}}$.

Analyzing the Given Problem: $\left(2 x^7-6\right) \div \left(x^5+4\right)$

Now, let's return to our original problem: ${(2x^7 - 6) \div (x^5 + 4)}$. To determine if synthetic division is appropriate, we need to assess whether the divisor, ${x^5 + 4}$, meets the requirements for synthetic division.

Evaluating the Divisor

The divisor in our problem is ${x^5 + 4}$. As we established earlier, synthetic division requires the divisor to be a linear binomial in the form ${x - c}$. In this case, the divisor is a quintic binomial (degree 5), not a linear one. This fundamental difference immediately tells us that synthetic division is not the appropriate method for this problem.

Why Synthetic Division Fails Here

Synthetic division works by systematically reducing the degree of the polynomial being divided based on the linear nature of the divisor. When the divisor is not linear, the mechanics of synthetic division break down. The algorithm relies on the simple relationship between the coefficients of the dividend and the constant term of the linear divisor to efficiently compute the quotient and remainder. This relationship does not hold when the divisor is of a higher degree.

Rewriting the Polynomials

Let's rewrite the polynomials in standard form, including any zero coefficients for missing terms. This will help illustrate why synthetic division is unsuitable:

Dividend: ${2x^7 + 0x^6 + 0x^5 + 0x^4 + 0x^3 + 0x^2 + 0x - 6}$ Divisor: ${x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 4}$

Notice that the divisor ${x^5 + 4}$ is far from the linear form ${x - c}$. Attempting to apply synthetic division here would lead to an incorrect and nonsensical result.

Alternative Methods for Polynomial Division

Since synthetic division is not suitable for dividing ${(2x^7 - 6)}$ by ${(x^5 + 4)}$, we need to consider alternative methods. The most common and versatile method for polynomial division when the divisor is not linear is polynomial long division.

Polynomial Long Division

Polynomial long division is a method that can handle divisors of any degree. It mirrors the long division process used for numbers but extends it to polynomials. Here’s a step-by-step overview of the process:

1. Set Up the Division: Write the dividend (the polynomial being divided) inside the division symbol and the divisor outside.
2. Divide the Leading Terms: Divide the leading term of the dividend by the leading term of the divisor. Write the result above the division symbol as part of the quotient.
3. Multiply: Multiply the entire divisor by the term you just wrote in the quotient. Write the result below the dividend, aligning terms with the same degree.
4. Subtract: Subtract the product from the dividend. Be careful to change the signs of the terms being subtracted.
5. Bring Down: Bring down the next term from the dividend and write it next to the result of the subtraction.
6. Repeat: Repeat steps 2-5 until all terms of the dividend have been used.
7. Interpret the Results: The polynomial written above the division symbol is the quotient, and the remaining polynomial after the last subtraction is the remainder.

Applying Polynomial Long Division to Our Problem

Let’s apply polynomial long division to ${(2x^7 - 6) \div (x^5 + 4)}$:

1. Set Up:

                 _____________
   

x^5 + 4 | 2x^7 + 0x^6 + 0x^5 + 0x^4 + 0x^3 + 0x^2 + 0x - 6 2. **Divide:** Divide ${2x^7}$ by ${x^5}$ to get ${2x^2}$. Write ${2x^2}$ above the division symbol. 2x^2__________ x^5 + 4 | 2x^7 + 0x^6 + 0x^5 + 0x^4 + 0x^3 + 0x^2 + 0x - 6 3. **Multiply:** Multiply ${2x^2}$ by ${x^5 + 4}$ to get ${2x^7 + 8x^2}$. Write this below the dividend. 2x^2__________ x^5 + 4 | 2x^7 + 0x^6 + 0x^5 + 0x^4 + 0x^3 + 0x^2 + 0x - 6 2x^7 + 0x^6 + 0x^5 + 0x^4 + 0x^3 + 8x^2 + 0x + 0 4. **Subtract:** Subtract ${2x^7 + 8x^2}$ from the dividend. 2x^2__________ x^5 + 4 | 2x^7 + 0x^6 + 0x^5 + 0x^4 + 0x^3 + 0x^2 + 0x - 6 - (2x^7 + 0x^6 + 0x^5 + 0x^4 + 0x^3 + 8x^2 + 0x + 0) ______________________________________________ -8x^2 - 6 ``` 5. Bring Down: There are no more terms to bring down, so we proceed to interpret the results. 6. Interpret: The quotient is ${2x^2}$, and the remainder is ${-8x^2 - 6}$.

Therefore, ${(2x^7 - 6) \div (x^5 + 4) = 2x^2 + \frac{-8x^2 - 6}{x^5 + 4}}$.

Other Division Techniques

While polynomial long division is the most universally applicable method, other techniques can be used in specific scenarios. For instance, if the divisor can be factored, we might use techniques like partial fraction decomposition or simplify the expression algebraically. However, these methods are not as generally applicable as polynomial long division.

Conclusion: Choosing the Right Method

In conclusion, the division problem ${(2x^7 - 6) \div (x^5 + 4)}$ is not a candidate for the synthetic division process. Synthetic division is limited to cases where the divisor is a linear binomial of the form ${x - c}$. Since the divisor in this problem, ${x^5 + 4}$, is a quintic binomial, we must use polynomial long division or other appropriate methods.

Understanding the requirements and limitations of different division techniques is crucial for efficient problem-solving in mathematics. By recognizing when synthetic division is applicable and when it is not, we can choose the right approach and avoid unnecessary complications. Polynomial long division, while more involved than synthetic division, provides a robust method for dividing polynomials with divisors of any degree.

Therefore, when faced with polynomial division, always consider the nature of the divisor before selecting a method. This ensures accuracy and efficiency in solving the problem.

Final Answer: No, synthetic division cannot be used for ${(2x^7 - 6) \div (x^5 + 4)}$ because the divisor ${x^5 + 4}$ is not a linear binomial.