Problem 33 - Algebra and Trigonometry

Question

Find the quotient and remainder using synthetic division. \(\frac{2 x^{3}+3 x^{2}-2 x+1}{x-\frac{1}{2}}\)

Step-by-Step Solution

Answer

The quotient is \(2x^2 + 4x\) and the remainder is 1.

Step 1: Identify the dividend and divisor

In this problem, the dividend is the polynomial \(2x^3 + 3x^2 - 2x + 1\) and the divisor is \(x - \frac{1}{2}\). We will perform synthetic division with these.

Step 2: Set up the synthetic division

List the coefficients of the polynomial: \(2, 3, -2, 1\). The root of the divisor \(x - \frac{1}{2}\) is \(\frac{1}{2}\). Use \(\frac{1}{2}\) on the left and the coefficients on the right.

Step 3: Perform the synthetic division calculation

1. Bring down the first coefficient \(2\) directly.2. Multiply \(2\) by \(\frac{1}{2}\) and write the result \(1\) under the next coefficient.3. Add \(3\) and \(1\) to get \(4\).4. Multiply \(4\) by \(\frac{1}{2}\) to get \(2\), and write it under the next coefficient.5. Add \(-2 + 2 = 0\).6. Multiply \(0\) by \(\frac{1}{2}\) to get \(0\), and write it under the last coefficient.7. Add \(1 + 0 = 1\).

Step 4: Interpret the result

The result of the synthetic division provides us with a quotient and a remainder. The numbers obtained are \(2, 4, 0\) with a remainder of \(1\). This implies the quotient polynomial is \(2x^2 + 4x + 0\), and the remainder is \(1\).

Key Concepts

Polynomial Division

Polynomial division is a method to divide one polynomial by another. In the exercise given, the division of the polynomial \( 2x^3 + 3x^2 - 2x + 1 \) by \( x - \frac{1}{2} \) is performed through synthetic division. This technique is a streamlined form of polynomial division when the divisor is of the first degree. Unlike long division, which might involve cumbersome steps, synthetic division simplifies the process by dealing primarily with the coefficients of the polynomial.

• Start by identifying the coefficients of the dividend polynomial as \( 2, 3, -2, \) and \( 1 \).
• The divisor \( x - \frac{1}{2} \) implies a synthetic factor, in this case, \( \frac{1}{2} \).

This synthetic factor is used to successively multiply and accumulate results through the coefficients to provide a quotient and remainder efficiently. This method is particularly useful for quick calculations in algebra and can aid in finding roots and factoring polynomials when the divisor is a linear binomial.

Quotient and Remainder

The quotient and remainder are the outcomes of the division process. In our problem, after applying synthetic division, the quotient is the polynomial \( 2x^2 + 4x \) and the remainder is \( 1 \). Understanding these terms helps make sense of the results:

• The quotient represents the polynomial obtained without the remainder after division, indicating how many times the divisor fits into the dividend.
• The remainder is what's "left over" after the division has been completed and can't be divided further by the divisor.

The complete division can be expressed as:\[ \frac{2x^3 + 3x^2 - 2x + 1}{x - \frac{1}{2}} = 2x^2 + 4x + \frac{1}{x - \frac{1}{2}} \]This demonstrates that the original polynomial can be broken down into "pieces" - one part that divides evenly and one small part that remains. Understanding both components is crucial for mastering polynomial division and comprehending polynomial behavior.

Algebra

Algebra is the broader branch of mathematics dealing with symbols and rules for manipulating those symbols. It provides a language for expressing mathematical relationships and concepts efficiently. Synthetic division is just one among many algebraic techniques designed to simplify and solve problems.

• Using symbols to represent numbers in equations allows for a comprehensive understanding and manipulation of quantities.
• Understanding algebraic techniques like polynomial division aids in solving more complex equations encountered in higher mathematics.

The practice of breaking down larger expressions, as seen in this division exercise, illustrates how algebra helps to decode intricate mathematical relationships, promoting clearer and more structured thinking. Mastery of these concepts equips students with tools to analyze patterns and solve problems in various fields, from science to engineering.