Synthetic Division Solution For (x^3 + 1) Divided By (x - 1)

Synthetic division is a streamlined method for dividing a polynomial by a linear divisor of the form x - a. It's a powerful shortcut compared to long division, especially when dealing with higher-degree polynomials. This method hinges on manipulating the coefficients of the polynomial and the constant term of the divisor to efficiently determine the quotient and remainder.

Before diving into the specific problem, it's crucial to understand the underlying principle of synthetic division. The process involves setting up a table with the coefficients of the dividend (the polynomial being divided) and the root of the divisor (the value of x that makes the divisor equal to zero). We then perform a series of multiplications and additions to arrive at the coefficients of the quotient and the remainder. This approach simplifies the division process by eliminating the need to write out the variables and exponents, focusing solely on the numerical relationships.

The beauty of synthetic division lies in its efficiency. By focusing on the coefficients, the process becomes much faster and less prone to errors compared to traditional long division. However, it's important to remember that synthetic division is specifically designed for dividing by linear expressions. When dividing by polynomials of higher degrees, long division is still the go-to method. Nonetheless, for linear divisors, synthetic division offers a significant advantage in terms of speed and simplicity. Synthetic division is a fundamental tool in polynomial algebra, especially useful in simplifying complex expressions and solving polynomial equations.

Let's apply synthetic division to the given problem: $(x^3 + 1) ÷ (x - 1)$. The first step is to identify the coefficients of the dividend, which is the polynomial $(x^3 + 1)$. We need to ensure that all powers of x are represented, even if their coefficients are zero. In this case, we have a term for $x^3$, but the terms for $x^2$ and x are missing. Therefore, we write the dividend as $1x^3 + 0x^2 + 0x + 1$. This gives us the coefficients 1, 0, 0, and 1. These coefficients are essential for setting up the synthetic division table correctly. Failing to include the zero coefficients can lead to incorrect results, as each coefficient's position is crucial in the synthetic division process.

Next, we need to find the root of the divisor, which is $(x - 1)$. To do this, we set $(x - 1)$ equal to zero and solve for x. This gives us $x = 1$. This value, 1, is the number we'll use in the synthetic division process. It represents the value that, when substituted into the divisor, makes it equal to zero. This root plays a central role in the synthetic division, as it is used in the multiplication steps to determine the coefficients of the quotient and the remainder. With the coefficients of the dividend and the root of the divisor identified, we are now ready to set up the synthetic division table and proceed with the calculations.

Now, let's perform the synthetic division. We set up the synthetic division table by writing the root of the divisor (which is 1) to the left, and the coefficients of the dividend (1, 0, 0, and 1) to the right. Draw a horizontal line below the coefficients, leaving space for the numbers we'll calculate during the process. The setup is crucial for maintaining the correct order of operations and ensuring accurate results. The first step in the actual division is to bring down the first coefficient (which is 1) below the line. This number will be the leading coefficient of our quotient.

Next, we multiply the root (1) by the number we just brought down (1), which gives us 1. We write this result under the second coefficient (0). Then, we add the second coefficient (0) and the result we just wrote (1), which gives us 1. This sum becomes the next coefficient in our quotient. We repeat this process: multiply the root (1) by the latest result (1), which gives us 1. Write this under the third coefficient (0). Add them together (0 + 1), which gives us 1. This becomes the next coefficient in our quotient. Finally, multiply the root (1) by the latest result (1), which gives us 1. Write this under the last coefficient (1). Add them together (1 + 1), which gives us 2. This last number is the remainder.

The numbers below the line, excluding the last one, represent the coefficients of the quotient. In this case, we have 1, 1, and 1. The last number, 2, is the remainder. Remember that the degree of the quotient is one less than the degree of the dividend. Since our dividend was a cubic polynomial (degree 3), the quotient will be a quadratic polynomial (degree 2). Therefore, the coefficients 1, 1, and 1 correspond to the terms $1x^2$, 1x, and 1, respectively. This step-by-step process, carefully executed, allows us to efficiently determine the quotient and remainder of the polynomial division.

Now, let's interpret the results of the synthetic division. We found the coefficients of the quotient to be 1, 1, and 1, and the remainder to be 2. As we discussed earlier, the coefficients 1, 1, and 1 correspond to the terms $1x^2$, x, and 1, respectively. Therefore, the quotient is $x^2 + x + 1$. The remainder, 2, is written as a fraction over the divisor, which is $(x - 1)$. So, the remainder term is $rac{2}{x - 1}$.

Combining the quotient and the remainder term, we get the final result of the division: $x^2 + x + 1 + rac{2}{x - 1}$. This means that when we divide $(x^3 + 1)$ by $(x - 1)$, we get a quotient of $x^2 + x + 1$ and a remainder of 2. The remainder term indicates that the division is not exact, and the polynomial $(x^3 + 1)$ is not perfectly divisible by $(x - 1)$. The ability to correctly interpret the results of synthetic division is crucial for solving polynomial equations and simplifying rational expressions. The quotient and remainder provide valuable information about the relationship between the dividend and the divisor, allowing us to further analyze and manipulate polynomial expressions. In this specific case, the quotient $x^2 + x + 1$ represents the polynomial that, when multiplied by $(x - 1)$, comes closest to the original polynomial $(x^3 + 1)$.

Therefore, the quotient of the division $(x^3 + 1) ÷ (x - 1)$ is $x^2 + x + 1 + rac{2}{x - 1}$, which corresponds to option C.

Final Answer: C