Question

use synthetic division to find ((x^{2}+4x - 5)div(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 of the divisor

For the divisor \(x - 1\), set \(x - 1 = 0\), so \(x = 1\). We will use this root in synthetic division. The coefficients of the dividend \(x^{2}+4x - 5\) are \(1\) (for \(x^{2}\)), \(4\) (for \(x\)), and \(- 5\) (the constant term).

Step2: Perform synthetic division

Set up the synthetic division as follows:
Bring down the first coefficient (\(1\)):
\[

$$\begin{array}{r|rrr} 1 & 1 & 4 & -5 \\ & & 1 & 5 \\ \hline & 1 & 5 & 0 \\ \end{array}$$

\]
Multiply the root (\(1\)) by the brought - down coefficient (\(1\)) and add to the next coefficient: \(1\times1 + 4=5\). Then multiply the root (\(1\)) by this new coefficient (\(5\)) and add to the last coefficient: \(1\times5+( - 5)=0\).

The numbers in the bottom row (excluding the last one) are the coefficients of the quotient polynomial \(q(x)\), and the last number is the remainder \(r\). So the quotient \(q(x)=x + 5\) and the remainder \(r = 0\), and the divisor \(d(x)=x - 1\).

Answer:

\(x + 5+\frac{0}{x - 1}\) (or simply \(x + 5\) since \(\frac{0}{x - 1}=0\))