Question

use synthetic division to simplify \\(\frac{7x^5 - 25x^4 + 26x^3 + 5x^2 - 19x - 14}{x - 2}\\). write your answer in the form \\(q(x) + \frac{r}{d(x)}), where \\(q(x)\\) is a polynomial, \\(r\\) is an integer, and \\(d(x)\\) is a linear polynomial. simplify any fractions.

Explanation:

Step1: Arrange coefficients

The dividend is \(7x^5 - 25x^4 + 26x^3 + 5x^2 - 19x - 14\), so coefficients are \(7, -25, 26, 5, -19, -14\). Divisor is \(x - 2\), so \(r = 2\).

Step2: Synthetic division setup

Set up synthetic division:
\[

$$\begin{array}{r|rrrrrr} 2 & 7 & -25 & 26 & 5 & -19 & -14 \\ & & 14 & -22 & 8 & 26 & 14 \\ \hline & 7 & -11 & 4 & 7 & 0 \\ \end{array}$$

\]

Step3: Determine quotient and remainder

The bottom row gives coefficients of quotient \(q(x)\): \(7x^4 - 11x^3 + 4x^2 + 7x + 7\), and remainder \(r = 0\).

Answer:

\(7x^4 - 11x^3 + 4x^2 + 7x + 7 + \frac{0}{x - 2}\) (or simplified as \(7x^4 - 11x^3 + 4x^2 + 7x + 7\) since \(\frac{0}{x - 2}=0\))