Question

find the quotient and remainder using synthetic long division for \\(\frac{x^3 + 8x^2 + 8x - 14}{x + 2}\\). give your answer in the form quotient\\(+ \frac{\text{remainder}}{\text{divisor}}\\)

Explanation:

Step1: Set up synthetic division

The divisor is \(x + 2\), so we use \(c=-2\). The coefficients of the dividend \(x^{3}+8x^{2}+8x - 14\) are \(1, 8, 8, -14\).
Set up the synthetic division as:
\[

$$\begin{array}{r|rrrr} -2 & 1 & 8 & 8 & -14 \\ & & -2 & -12 & 8 \\ \hline & 1 & 6 & -4 & -6 \\ \end{array}$$

\]

Step2: Interpret the results

The last number in the bottom row is the remainder, which is \(-6\). The other numbers are the coefficients of the quotient polynomial. Since the dividend is a cubic polynomial, the quotient is a quadratic polynomial: \(x^{2}+6x - 4\).

Step3: Write the final form

Using the form \(quotient+\frac{remainder}{divisor}\), we substitute the quotient, remainder, and divisor (\(x + 2\)):
\(x^{2}+6x - 4+\frac{-6}{x + 2}\) or \(x^{2}+6x - 4-\frac{6}{x + 2}\)

Answer:

\(x^{2}+6x - 4-\frac{6}{x + 2}\)