Problem 23 - Basic Technical Mathematics with Calculus

Question

Perform the indicated divisions by synthetic division. $$\left(x^{3}+2 x^{2}-x-2\right) \div(x-1)$$

Step-by-Step Solution

Answer

The quotient is \( x^2 + 3x + 2 \) with a remainder of 0.

Step 1: Setup Synthetic Division

Write down the coefficients of the dividend polynomial \( x^3 + 2x^2 - x - 2 \) which are \([1, 2, -1, -2]\). Then, write the root of the divisor \( x-1 \), which is 1, on the left.

Step 2: Synthetic Division First Iteration

Bring down the leading coefficient (1) directly below the line. Multiply it by the root (1) and place the result (1) under the next coefficient (2). Add these to get the new coefficient (3).

Step 3: Synthetic Division Second Iteration

Take the new coefficient (3) and multiply it by the root (1), placing the result (3) under the next coefficient (-1). Add these to get the new coefficient (2).

Step 4: Synthetic Division Third Iteration

Take the new coefficient (2) and multiply it by the root (1), placing the result (2) under the next coefficient (-2). Add these to get the remainder (0).

Step 5: Write the Quotient and Remainder

The bottom row gives the coefficients of the quotient polynomial. Therefore, the quotient is \( x^2 + 3x + 2 \) with a remainder of 0.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. In this exercise, you're asked to divide the polynomial \(x^3 + 2x^2 - x - 2\) by \(x - 1\). Synthetic division is a special technique for dividing a polynomial by a binomial of the form \(x - c\). It simplifies the calculation compared to long division by focusing only on the essential coefficients and root of the divisor.
This process begins by identifying and listing the coefficients of the dividend polynomial, which in this case are \([1, 2, -1, -2]\). After identifying the coefficients, you also need to determine the root of the divisor, which is the value that makes \(x - 1 = 0\), and that root is 1.

Remainder Theorem

The Remainder Theorem states that if you divide a polynomial \(f(x)\) by \(x - c\), the remainder of that division is \(f(c)\). Essentially, this means evaluating the polynomial at \(c\) will give you the remainder. In synthetic division, because you're simplifying the process of obtaining the remainder and the quotient, it's easy to directly see if the divisor is a factor if the remainder is zero.
In the exercise provided, by using synthetic division, the calculation resulted in a remainder of zero, confirming that \(x - 1\) is indeed a factor of \(x^3 + 2x^2 - x - 2\). This conclusion is critical when analyzing polynomial behavior, especially when considering roots and possible factors.

Divisor Root

When performing synthetic division, a crucial step is identifying the root of the divisor. For a divisor \(x - c\), the root is simply \(c\). This root is used throughout the synthetic division process. In this case, the divisor given is \(x - 1\). Its root is 1 because substituting 1 into \(x - 1\) yields zero.
This root is used in each step of synthetic division to multiply and determine subsequent coefficients as you work through the polynomial. It's important because using the root allows you to simplify the division process while maintaining correctness. Understanding this concept helps in comprehending how and why synthetic division is effective and efficient.

Coefficient Calculation

Coefficient calculation is fundamental in synthetic division. Once you've set up the division framework with the coefficients of the polynomial and determined the root of the divisor, you embark on a series of multiplications and additions.
The process begins with bringing down the leading coefficient unaltered. For each subsequent coefficient, the operations involve multiplying the latest result by the root and adding it to the next coefficient in line. This iterative approach updates the coefficients step by step, forming the new coefficients of the result, which eventually outline the quotient polynomial.
In our example, starting with the coefficients \([1, 2, -1, -2]\), the synthetic division transforms them into new coefficients \([1, 3, 2]\), representing the quotient \(x^2 + 3x + 2\). This efficient calculation sequence underlines why understanding coefficient manipulation is crucial in polynomial division.