Question

use synthetic division to simplify \\(\frac{x^3 - 20x^2 - x + 20}{x - 1}\\). 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: Identify the root for synthetic division

For the divisor \(x - 1\), the root \(r = 1\). The coefficients of the dividend \(x^{3}-20x^{2}-x + 20\) are \(1\), \(-20\), \(-1\), \(20\).

Step2: Set up synthetic division

Write the root \(1\) and the coefficients:
\[

$$\begin{array}{r|rrrr} 1 & 1 & -20 & -1 & 20 \\ & & 1 & -19 & -20 \\ \hline & 1 & -19 & -20 & 0 \\ \end{array}$$

\]
The last number is the remainder \(r = 0\), and the other numbers are the coefficients of the quotient polynomial \(q(x)\). So \(q(x)=x^{2}-19x - 20\) and \(r = 0\), \(d(x)=x - 1\).

Step3: Write the result

Using the form \(q(x)+\frac{r}{d(x)}\), we substitute the values: \(x^{2}-19x - 20+\frac{0}{x - 1}\), which simplifies to \(x^{2}-19x - 20\) (since \(\frac{0}{x - 1}=0\)).

Answer:

\(x^{2}-19x - 20\)