Synthetic Division Explained Solve (x^4-1) ÷ (x-1)

In the realm of polynomial algebra, synthetic division stands out as a streamlined method for dividing a polynomial by a linear divisor. This technique is particularly useful for simplifying complex polynomial expressions and identifying roots. In this comprehensive guide, we will delve into the step-by-step process of using synthetic division to solve the problem $(x^4 - 1) ÷ (x - 1)$. We will not only demonstrate the mechanics of the method but also provide a deeper understanding of why it works and its applications in various mathematical contexts.

What is Synthetic Division?

Synthetic division is a shorthand method of polynomial division that is particularly useful when dividing by a linear factor of the form x - k. This method simplifies the long division process by focusing on the coefficients of the polynomials, thereby reducing the amount of writing and potential for errors. It's an efficient way to find the quotient and remainder when dividing a polynomial by a linear expression. Before we dive into the specifics of our problem, let's briefly review the traditional method of polynomial long division to appreciate the efficiency of synthetic division.

Traditional Polynomial Long Division: A Brief Overview

Polynomial long division is similar to the long division of numbers, but instead of digits, we're dealing with terms involving variables and exponents. The process involves dividing the highest degree term of the dividend (the polynomial being divided) by the highest degree term of the divisor (the polynomial we're dividing by), multiplying the result by the divisor, subtracting it from the dividend, and then bringing down the next term. This process repeats until we can no longer divide. While effective, long division can be cumbersome, especially for higher-degree polynomials. This is where synthetic division shines, offering a more streamlined approach.

Setting Up the Synthetic Division

To set up the synthetic division for $(x^4 - 1) ÷ (x - 1)$, we first identify the coefficients of the dividend polynomial and the root of the divisor. The dividend is $x^4 - 1$, which can be rewritten as $x^4 + 0x^3 + 0x^2 + 0x - 1$ to explicitly show all the coefficients. The coefficients are therefore 1, 0, 0, 0, and -1. The divisor is $(x - 1)$, so we take the root, which is k = 1. This value will be placed outside the division symbol in the synthetic division setup. Now, we'll arrange these elements in the synthetic division format. Write the root of the divisor (1 in this case) to the left, and then list the coefficients of the dividend (1, 0, 0, 0, -1) across the top row. Make sure to include placeholders (zeros) for any missing terms in the polynomial. This is crucial for maintaining the correct place value during the division process.

Step-by-Step Process of Synthetic Division for (x^4 - 1) ÷ (x - 1)

The synthetic division process is a methodical sequence of steps that makes polynomial division more manageable. Let's break down each step in the context of our problem, (x^4 - 1) ÷ (x - 1), to ensure clarity and understanding.

1. Bring Down the First Coefficient: The first step is to bring down the leading coefficient of the dividend (which is 1 in our case) below the horizontal line. This will be the first coefficient of our quotient.
2. Multiply and Add: Next, multiply the value we just brought down (1) by the root of the divisor (1). The result (1 * 1 = 1) is placed under the next coefficient of the dividend (which is 0). Then, add these two numbers together (0 + 1 = 1) and write the sum below the line. This sum is the second coefficient of our quotient.
3. Repeat the Process: Continue this process of multiplying the last number below the line by the root (1) and adding the result to the next coefficient of the dividend. So, multiply 1 (the last number below the line) by 1 (the root) to get 1. Place this under the next coefficient (0), and add them together (0 + 1 = 1). Write the sum (1) below the line. This will be the third coefficient of our quotient.
4. Final Iteration: Repeat this process one last time. Multiply 1 (the last number below the line) by 1 (the root) to get 1. Place this under the final coefficient (-1), and add them together (-1 + 1 = 0). Write the sum (0) below the line. This final number represents the remainder.

By following these steps, we efficiently transform the original division problem into a series of simple multiplications and additions. This process not only simplifies the calculation but also provides a clear path to finding both the quotient and the remainder of the division.

Interpreting the Results

After completing the synthetic division, the numbers below the line represent the coefficients of the quotient and the remainder. In our case, the numbers below the line are 1, 1, 1, and 0. The last number (0) is the remainder, and the other numbers are the coefficients of the quotient polynomial. Since we started with a polynomial of degree 4 and divided by a polynomial of degree 1, the quotient will be a polynomial of degree 3. Therefore, the coefficients 1, 1, and 1 correspond to the terms $x^3$, $x^2$, and x, respectively. The quotient is thus $1x^3 + 1x^2 + 1x + 1$, which simplifies to $x^3 + x^2 + x + 1$. The remainder is 0, indicating that $(x - 1)$ divides evenly into $(x^4 - 1)$.

Understanding the Quotient and Remainder

The quotient, $x^3 + x^2 + x + 1$, represents the result of the division, while the remainder (0 in this case) indicates the amount left over after the division. A remainder of 0 is particularly significant because it tells us that the divisor is a factor of the dividend. In other words, $(x - 1)$ is a factor of $(x^4 - 1)$. This is a crucial piece of information when factoring polynomials or solving polynomial equations. The ability to quickly determine factors and roots is one of the key advantages of using synthetic division.

The Quotient: The Final Answer

Therefore, the quotient of $(x^4 - 1) ÷ (x - 1)$ is $x^3 + x^2 + x + 1$. This corresponds to option C in the given choices. Synthetic division has provided us with a straightforward method to arrive at this solution, bypassing the complexities of traditional long division. The ability to accurately and efficiently perform polynomial division is essential for various mathematical applications, including solving equations, simplifying expressions, and analyzing the behavior of polynomial functions.

Why Synthetic Division Works: A Deeper Look

To fully appreciate synthetic division, it’s important to understand why it works. The method is essentially a condensed form of polynomial long division, but instead of writing out the variables and exponents, we work solely with the coefficients. This simplification is possible because the place value of each term is maintained throughout the process. The multiplication and addition steps in synthetic division mirror the steps of polynomial long division, but in a more streamlined format. By focusing on the numerical relationships between the coefficients, synthetic division reduces the cognitive load and minimizes the chances of making errors.

Applications of Synthetic Division

Synthetic division is not just a mathematical trick; it's a powerful tool with several important applications in algebra and calculus. Some of the key applications include:

Finding Roots of Polynomials

As demonstrated in our problem, synthetic division can be used to determine if a linear expression is a factor of a polynomial. If the remainder is 0, then the root of the divisor is a root of the polynomial. This is particularly useful for finding rational roots of higher-degree polynomials. By testing potential roots using synthetic division, we can systematically narrow down the possibilities and find the actual roots.

Factoring Polynomials

When we find a root using synthetic division, we also obtain the quotient polynomial. This quotient is a lower-degree polynomial that can be further factored. By repeatedly applying synthetic division, we can break down a complex polynomial into its linear factors. This is essential for solving polynomial equations and analyzing the behavior of polynomial functions.

Evaluating Polynomials

Synthetic division can also be used to evaluate a polynomial at a specific value. This is known as the Remainder Theorem, which states that if a polynomial f(x) is divided by (x - k), then the remainder is equal to f(k). This provides an efficient way to calculate the value of a polynomial at a given point without direct substitution.

Simplifying Rational Expressions

Synthetic division can be used to simplify rational expressions by dividing the numerator by the denominator. This can help to identify any common factors and reduce the expression to its simplest form. Simplifying rational expressions is a fundamental skill in algebra and calculus, and synthetic division provides a valuable tool for this task.

Conclusion

In conclusion, synthetic division is a highly efficient and valuable technique for dividing polynomials, particularly by linear divisors. It simplifies the division process, making it less prone to errors and more manageable. Through the step-by-step solution of $(x^4 - 1) ÷ (x - 1)$, we've demonstrated the practical application of synthetic division and its ability to quickly yield the quotient $x^3 + x^2 + x + 1$. Moreover, we’ve explored the broader implications of synthetic division, from finding roots and factoring polynomials to evaluating polynomials and simplifying rational expressions. By mastering synthetic division, students and mathematicians alike gain a powerful tool for tackling a wide range of algebraic problems. The ability to manipulate and simplify polynomial expressions is fundamental to success in advanced mathematics, and synthetic division provides a crucial pathway to achieving this mastery.