Problem 23 - Precalculus with Calculus Previews

Question

In Problems \(23-32\), use synthetic division to find the quotient \(q(x)\) and remainder \(r\) when \(f(x)\) is divided by the given linear polynomial. $$ f(x)=2 x^{2}-x+5 ; x-2 $$

Step-by-Step Solution

Answer

The quotient is \(q(x) = 2x + 3\) and the remainder is \(r = 11\).

Step 1: Set Up the Synthetic Division

For synthetic division, take the zero of the divisor. Since the divisor is \(x - 2\), the zero is \(2\). Write this zero on the left side. On the right, list the coefficients of \(f(x)\): \(2, -1, 5\). Prepare space below for carrying out the division steps.

Step 2: Bring Down the Leading Coefficient

Bring down the leading coefficient of \(f(x)\), which is \(2\), directly below the division line. This is the beginning of our quotient \(q(x)\).

Step 3: Multiply and Add

Multiply the number just written below the line (\(2\)) by the zero of the divisor (\(2\)), and write the result \(4\) underneath the next coefficient (\(-1\)). Then, add \(-1\) and \(4\) to get \(3\), and write this under the line.

Step 4: Repeat Multiply and Add Step

Repeat the multiply and add process: Multiply the number you just wrote (\(3\)) by \(2\) to get \(6\). Write \(6\) under the next coefficient (\(5\)). Add these numbers to get \(11\), and write it under the line as the remainder \(r\).

Step 5: Write the Quotient and Remainder

The numbers below the line, now \(2\) and \(3\), represent the coefficients of the quotient polynomial \(q(x)\). Therefore, the quotient is \(2x + 3\) and the remainder is \(11\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, especially when the divisor is a simple linear polynomial. It is much like long division with numbers and helps in simplifying expressions and solving equations. There are two main methods for polynomial division: long division and synthetic division. Long division is a more straightforward approach, similar to the standard arithmetic division process. However, when dividing by a linear divisor, synthetic division is a quick and efficient method. In synthetic division, the main idea is to work with the coefficients of the polynomial terms rather than the entire expression. This reduces the complexity of the division process and makes it faster. It streamlines calculations by focusing on the roots and coefficients, thus making it an essential tool in algebra.

Quotient and Remainder

When dividing polynomials, you often end up with two results: the quotient and the remainder. The division process will yield a quotient polynomial, which is the result of dividing the original polynomial by the divisor, up to a certain degree, and a remainder, which represents what is left after the division is complete.The quotient refers to the polynomial that you get as the main part of your division result, excluding any leftover part. For example, if you divide a quadratic polynomial by a linear one, your quotient will generally be a linear polynomial.The remainder is the part that cannot be included in the quotient because it is of a lower degree than the divisor. In any polynomial division, we express the original polynomial as:\[ f(x) = (x - a)q(x) + r \]where \( r \) is the remainder. In synthetic division, it is crucial to pay attention to the final position of the remainder in the line of results.

Polynomial Coefficients

Polynomial coefficients are the numerical factors that multiply each term of a polynomial. For a polynomial \( f(x) = ax^n + bx^{n-1} + \,\ldots\, + k \), the coefficients are \( a, b, \,\dots\,, k \). Understanding these coefficients is crucial in synthetic division, as it is these numbers that form the foundation of the operations.In synthetic division, you start by writing down the coefficients of the polynomial you wish to divide. This list of coefficients represents the polynomial's terms in decreasing order of degree. For instance, with the polynomial \( 2x^2 - x + 5 \), the coefficients are \( 2, -1, 5 \).By utilizing these coefficients, synthetic division divides the polynomial systematically, allowing us to discover both the quotient and remainder quickly. This reliance on coefficients in synthetic division demonstrates the simplified nature of the process, focusing on numerical operations rather than algebraic manipulations.