Problem 60 - Algebra for College Students

Question

Use synthetic division to determine the quotient and remainder. $$ \left(x^{3}+6 x^{2}-5 x-1\right) \div(x-1) $$

Step-by-Step Solution

Answer

The quotient is \(x^2 + 7x + 2\) with a remainder of 1.

Step 1: Identify Coefficients and Divisor

The polynomial to be divided is \(x^3 + 6x^2 - 5x - 1\). Identify its coefficients: 1, 6, -5, and -1. The divisor is \(x-1\), so the root for synthetic division is 1.

Step 2: Set Up Synthetic Division

Write down the coefficients of the polynomial: 1, 6, -5, -1. Next to these, draw a half-box to separate the work area from the coefficients and place the root of the divisor, 1, to the left.

Step 3: Begin Synthetic Division

Start the synthetic division by bringing down the first coefficient (1) to the bottom row, as it is. This is the beginning of our quotient.

Step 4: Multiply and Add

Multiply the number you just brought down (1) by the divisor root (1). Write this product (1) below the next coefficient (6). Add this product to the next coefficient (6) to get 7. Write this sum on the bottom row.

Step 5: Repeat Multiply and Add

Continue the process: multiply the new bottom number (7) by the divisor root (1), and write the product (7) below the next coefficient (-5). Add to get 2. Write this on the bottom row.

Step 6: Multiply and Add for Final Calculation

Multiply the bottom number (2) by the divisor root (1) and write the product (2) below the next coefficient (-1). Add this to the next coefficient (-1) to get 1. This is your remainder.

Step 7: Write the Quotient and Remainder

The result from the bottom row, 1, 7, 2, represents the coefficients of the quotient \(x^2 + 7x + 2\). The last number (1) is the remainder. So, \((x^3 + 6x^2 - 5x - 1) \div (x-1) = x^2 + 7x + 2\) with a remainder of 1.

Key Concepts

polynomial division

Polynomial division is a method used to divide one polynomial by another. It is similar to long division, but with polynomials! When dealing with polynomials, we often use either long division or synthetic division to simplify the division process. In this article, we will focus on synthetic division, a streamlined way to divide polynomials that is particularly useful when the divisor is in the form of \(x - c\). Here, \(c\) is a constant. This method is quicker and involves fewer calculations.Synthetic division starts by setting up a work area that combines some elements of traditional division with the specific properties of polynomial coefficients. The process results in a quotient and possibly a remainder. It is important to note that the dividend (the polynomial being divided) must have descending powers of \(x\) for synthetic division to be applied effectively. If any terms are missing, it is necessary to include them with coefficients of zero.

quotient and remainder

In the realm of polynomial division, the quotient and remainder play crucial roles. During the division process, the quotient is the result from dividing the dividend (the main polynomial) by the divisor (usually expressed as \(x - c\)). The remainder represents what is left after the division is performed.For synthetic division, it is typical to express the results as:

• Quotient: The remaining numbers after performing the division correspond to the coefficients of the quotient polynomial.
• Remainder: The last number obtained at the end of the division process represents the remainder.

So, if the dividend is divided perfectly by the divisor, the remainder is zero. However, it is common to have a non-zero remainder, as seen in the exercise where a remainder of 1 was obtained. The final expression combines both the quotient polynomial and the remainder to accurately represent the division outcome.

dividing polynomials

Dividing polynomials involves determining how one polynomial, referred to as the dividend, can be broken down by another, called the divisor. The division process aims to express the dividend as the product of the divisor and a quotient polynomial, plus a remainder.Synthetic division simplifies this by using only the coefficients of the polynomials, streamlining the calculation process. This method efficiently handles polynomials divided by linear divisors of the form \(x - c\). Here's a quick overview of how synthetic division works:

• List the coefficients of the polynomial to be divided.
• Use the opposite of the constant term from the divisor \(x - c\) as the root.
• Perform synthetic division by multiplying and adding across the "work line" to get the quotient coefficients and remainder.

Despite its efficiency, synthetic division mainly applies in specific cases where the divisor is linear. For more complex divisors, polynomial long division would be required, ensuring any divisor can be handled. This approach is useful for students learning to manipulate and divide polynomials with ease.